Английская Википедия:Binomial theorem

Шаблон:Short description Шаблон:Image frame In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial Шаблон:Math into a sum involving terms of the form Шаблон:Math, where the exponents Шаблон:Mvar and Шаблон:Mvar are nonnegative integers with Шаблон:Math, and the coefficient Шаблон:Mvar of each term is a specific positive integer depending on Шаблон:Mvar and Шаблон:Mvar. For example, for Шаблон:Math, <math display="block">(x+y)^4 = x^4 + 4 x^3y + 6 x^2 y^2 + 4 x y^3 + y^4. </math>

The coefficient Шаблон:Mvar in the term of Шаблон:Math is known as the binomial coefficient <math>\tbinom{n}{b}</math> or <math>\tbinom{n}{c}</math> (the two have the same value). These coefficients for varying Шаблон:Mvar and Шаблон:Mvar can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where <math>\tbinom{n}{b}</math> gives the number of different combinations (i.e. subsets) of Шаблон:Mvar elements that can be chosen from an Шаблон:Mvar-element set. Therefore <math>\tbinom{n}{b}</math> is usually pronounced as "Шаблон:Mvar choose Шаблон:Mvar".

History

Special cases of the binomial theorem were known since at least the 4th century BC when Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent <math>n=2</math>.^[1] Greek mathematician Diophantus cubed various binomials, including <math>x-1</math>.^[1] Indian mathematician Aryabhata's method for finding cube roots, from around 510 CE, suggests that he knew the binomial formula for exponent <math>n=3</math>.^[1]

Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting Шаблон:Mvar objects out of Шаблон:Mvar without replacement, were of interest to ancient Indian mathematicians. The earliest known reference to this combinatorial problem is the Chandaḥśāstra by the Indian lyricist Pingala (c. 200 BC), which contains a method for its solution.^[2]Шаблон:Rp The commentator Halayudha from the 10th century AD explains this method.^[2]Шаблон:Pn By the 6th century AD, the Indian mathematicians probably knew how to express this as a quotient <math display="inline">\frac{n!}{(n-k)!k!}</math>,^[3] and a clear statement of this rule can be found in the 12th century text Lilavati by Bhaskara.^[3]

The first known formulation of the binomial theorem and the table of binomial coefficients appears in a work by Al-Karaji, quoted by Al-Samaw'al in his "al-Bahir".^[4]^[5]^[6] Al-Karaji described the triangular pattern of the binomial coefficients^[7] and also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using an early form of mathematical induction.^[7] The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, although many of his mathematical works are lost.^[1] The binomial expansions of small degrees were known in the 13th century mathematical works of Yang Hui^[8] and also Chu Shih-Chieh.^[1] Yang Hui attributes the method to a much earlier 11th century text of Jia Xian, although those writings are now also lost.^[2]Шаблон:Rp

In 1544, Michael Stifel introduced the term "binomial coefficient" and showed how to use them to express <math>(1+x)^n</math> in terms of <math>(1+x)^{n-1}</math>, via "Pascal's triangle".^[9] Blaise Pascal studied the eponymous triangle comprehensively in his Traité du triangle arithmétique.^[10] However, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, Niccolò Fontana Tartaglia, and Simon Stevin.^[9]

Isaac Newton is generally credited with discovering the generalized binomial theorem, valid for any real exponent, in 1665.^[9]^[11] It was discovered independently in 1670 by James Gregory.^[12]

Statement

According to the theorem, the expansion of any nonnegative integer power Шаблон:Mvar of the binomial Шаблон:Math is a sum of the form <math display="block">(x+y)^n = {n \choose 0}x^n y^0 + {n \choose 1}x^{n-1} y^1 + {n \choose 2}x^{n-2} y^2 + \cdots + {n \choose n-1}x^1 y^{n-1} + {n \choose n}x^0 y^n,</math> where each <math> \tbinom nk </math> is a positive integer known as a binomial coefficient, defined as

<math display=block>\binom nk = \frac{n!}{k!\,(n-k)!} = \frac{n(n-1)(n-2)\cdots(n-k + 1)}{k(k-1)(k-2)\cdots2\cdot1}.</math>

This formula is also referred to as the binomial formula or the binomial identity. Using summation notation, it can be written more concisely as <math display="block">(x+y)^n = \sum_{k=0}^n {n \choose k}x^{n-k}y^k = \sum_{k=0}^n {n \choose k}x^{k}y^{n-k}.</math>

The final expression follows from the previous one by the symmetry of Шаблон:Mvar and Шаблон:Mvar in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetrical, <math display=inline>\binom nk = \binom n{n-k}.</math>

A simple variant of the binomial formula is obtained by substituting Шаблон:Math for Шаблон:Mvar, so that it involves only a single variable. In this form, the formula reads <math display=block>\begin{align} (1+x)^n &= {n \choose 0}x^0 + {n \choose 1}x^1 + {n \choose 2}x^2 + \cdots + {n \choose {n-1}}x^{n-1} + {n \choose n}x^n \\[4mu] &= \sum_{k=0}^n {n \choose k}x^k. \vphantom{\Bigg)} \end{align}</math>

Examples

Here are the first few cases of the binomial theorem: <math display="block">\begin{align} (x+y)^0 & = 1, \\[8pt] (x+y)^1 & = x + y, \\[8pt] (x+y)^2 & = x^2 + 2xy + y^2, \\[8pt] (x+y)^3 & = x^3 + 3x^2y + 3xy^2 + y^3, \\[8pt] (x+y)^4 & = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4, \\[8pt] (x+y)^5 & = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5, \\[8pt] (x+y)^6 & = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6, \\[8pt] (x+y)^7 & = x^7 + 7x^6y + 21x^5y^2 + 35x^4y^3 + 35x^3y^4 + 21x^2y^5 + 7xy^6 + y^7, \\[8pt] (x+y)^8 & = x^8 + 8x^7y + 28x^6y^2 + 56x^5y^3 + 70x^4y^4 + 56x^3y^5 + 28x^2y^6 + 8xy^7 + y^8. \end{align}</math> In general, for the expansion of Шаблон:Math on the right side in the Шаблон:Mvarth row (numbered so that the top row is the 0th row):

the exponents of Шаблон:Mvar in the terms are Шаблон:Math (the last term implicitly contains Шаблон:Math);
the exponents of Шаблон:Mvar in the terms are Шаблон:Math (the first term implicitly contains Шаблон:Math);
the coefficients form the Шаблон:Mvarth row of Pascal's triangle;
before combining like terms, there are Шаблон:Math terms Шаблон:Math in the expansion (not shown);
after combining like terms, there are Шаблон:Math terms, and their coefficients sum to Шаблон:Math.

An example illustrating the last two points: <math display="block">\begin{align} (x+y)^3 & = xxx + xxy + xyx + xyy + yxx + yxy + yyx + yyy & (2^3 \text{ terms}) \\

       & = x^3 + 3x^2y + 3xy^2 + y^3 & (3 + 1 \text{ terms})

\end{align}</math> with <math>1 + 3 + 3 + 1 = 2^3</math>.

A simple example with a specific positive value of Шаблон:Math: <math display="block">\begin{align} (x+2)^3 &= x^3 + 3x^2(2) + 3x(2)^2 + 2^3 \\ &= x^3 + 6x^2 + 12x + 8. \end{align}</math>

A simple example with a specific negative value of Шаблон:Math: <math display="block">\begin{align} (x-2)^3 &= x^3 - 3x^2(2) + 3x(2)^2 - 2^3 \\ &= x^3 - 6x^2 + 12x - 8. \end{align}</math>

Geometric explanation

Файл:Binomial theorem visualisation.svg

Visualisation of binomial expansion up to the 4th power

For positive values of Шаблон:Mvar and Шаблон:Mvar, the binomial theorem with Шаблон:Math is the geometrically evident fact that a square of side Шаблон:Math can be cut into a square of side Шаблон:Mvar, a square of side Шаблон:Mvar, and two rectangles with sides Шаблон:Mvar and Шаблон:Mvar. With Шаблон:Math, the theorem states that a cube of side Шаблон:Math can be cut into a cube of side Шаблон:Mvar, a cube of side Шаблон:Mvar, three Шаблон:Math rectangular boxes, and three Шаблон:Math rectangular boxes.

In calculus, this picture also gives a geometric proof of the derivative <math>(x^n)'=nx^{n-1}:</math>^[13] if one sets <math>a=x</math> and <math>b=\Delta x,</math> interpreting Шаблон:Mvar as an infinitesimal change in Шаблон:Mvar, then this picture shows the infinitesimal change in the volume of an Шаблон:Mvar-dimensional hypercube, <math>(x+\Delta x)^n,</math> where the coefficient of the linear term (in <math>\Delta x</math>) is <math>nx^{n-1},</math> the area of the Шаблон:Mvar faces, each of dimension Шаблон:Math: <math display="block">(x+\Delta x)^n = x^n + nx^{n-1}\Delta x + \binom{n}{2}x^{n-2}(\Delta x)^2 + \cdots.</math> Substituting this into the definition of the derivative via a difference quotient and taking limits means that the higher order terms, <math>(\Delta x)^2</math> and higher, become negligible, and yields the formula <math>(x^n)'=nx^{n-1},</math> interpreted as

"the infinitesimal rate of change in volume of an Шаблон:Mvar-cube as side length varies is the area of Шаблон:Mvar of its Шаблон:Math-dimensional faces".

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral <math>\textstyle{\int x^{n-1}\,dx = \tfrac{1}{n} x^n}</math> – see proof of Cavalieri's quadrature formula for details.^[13]

Binomial coefficients

Шаблон:Main The coefficients that appear in the binomial expansion are called binomial coefficients. These are usually written <math>\tbinom{n}{k},</math> and pronounced "Шаблон:Mvar choose Шаблон:Mvar".

Formulas

The coefficient of Шаблон:Math is given by the formula <math display="block">\binom{n}{k} = \frac{n!}{k! \; (n-k)!},</math> which is defined in terms of the factorial function Шаблон:Math. Equivalently, this formula can be written <math display="block">\binom{n}{k} = \frac{n (n-1) \cdots (n-k+1)}{k (k-1) \cdots 1} = \prod_{\ell=1}^k \frac{n-\ell+1}{\ell} = \prod_{\ell=0}^{k-1} \frac{n-\ell}{k - \ell}</math> with Шаблон:Mvar factors in both the numerator and denominator of the fraction. Although this formula involves a fraction, the binomial coefficient <math>\tbinom{n}{k}</math> is actually an integer.

Combinatorial interpretation

The binomial coefficient <math> \tbinom nk </math> can be interpreted as the number of ways to choose Шаблон:Mvar elements from an Шаблон:Mvar-element set. This is related to binomials for the following reason: if we write Шаблон:Math as a product <math display="block">(x+y)(x+y)(x+y)\cdots(x+y),</math> then, according to the distributive law, there will be one term in the expansion for each choice of either Шаблон:Mvar or Шаблон:Mvar from each of the binomials of the product. For example, there will only be one term Шаблон:Math, corresponding to choosing Шаблон:Mvar from each binomial. However, there will be several terms of the form Шаблон:Math, one for each way of choosing exactly two binomials to contribute a Шаблон:Mvar. Therefore, after combining like terms, the coefficient of Шаблон:Math will be equal to the number of ways to choose exactly Шаблон:Math elements from an Шаблон:Mvar-element set.

Proofs

Combinatorial proof

Example

The coefficient of Шаблон:Math in <math display="block">\begin{align}

  (x+y)^3 &= (x+y)(x+y)(x+y) \\
  &= xxx + xxy + xyx + \underline{xyy} + yxx + \underline{yxy} + \underline{yyx} + yyy \\
  &= x^3 + 3x^2y + \underline{3xy^2} + y^3

\end{align}</math> equals <math>\tbinom{3}{2}=3</math> because there are three Шаблон:Math strings of length 3 with exactly two Шаблон:Mvars, namely, <math display="block">xyy, \; yxy, \; yyx,</math> corresponding to the three 2-element subsets of Шаблон:Math, namely, <math display="block">\{2,3\},\;\{1,3\},\;\{1,2\}, </math> where each subset specifies the positions of the Шаблон:Mvar in a corresponding string.

General case

Expanding Шаблон:Math yields the sum of the Шаблон:Math products of the form Шаблон:Math where each Шаблон:Math is Шаблон:Mvar or Шаблон:Mvar. Rearranging factors shows that each product equals Шаблон:Math for some Шаблон:Mvar between Шаблон:Math and Шаблон:Mvar. For a given Шаблон:Mvar, the following are proved equal in succession:

the number of terms equal to Шаблон:Math in the expansion
the number of Шаблон:Mvar-character Шаблон:Math strings having Шаблон:Mvar in exactly Шаблон:Mvar positions
the number of Шаблон:Mvar-element subsets of Шаблон:Math
<math>\tbinom{n}{k},</math> either by definition, or by a short combinatorial argument if one is defining <math>\tbinom{n}{k}</math> as <math>\tfrac{n!}{k! (n-k)!}.</math>

This proves the binomial theorem.

Inductive proof

Induction yields another proof of the binomial theorem. When Шаблон:Math, both sides equal Шаблон:Math, since Шаблон:Math and <math>\tbinom{0}{0}=1.</math> Now suppose that the equality holds for a given Шаблон:Mvar; we will prove it for Шаблон:Math. For Шаблон:Math, let Шаблон:Math denote the coefficient of Шаблон:Math in the polynomial Шаблон:Math. By the inductive hypothesis, Шаблон:Math is a polynomial in Шаблон:Mvar and Шаблон:Mvar such that Шаблон:Math is <math>\tbinom{n}{k}</math> if Шаблон:Math, and Шаблон:Mvar otherwise. The identity <math display="block"> (x+y)^{n+1} = x(x+y)^n + y(x+y)^n</math> shows that Шаблон:Math is also a polynomial in Шаблон:Mvar and Шаблон:Mvar, and <math display="block"> [(x+y)^{n+1}]_{j,k} = [(x+y)^n]_{j-1,k} + [(x+y)^n]_{j,k-1},</math> since if Шаблон:Math, then Шаблон:Math and Шаблон:Math. Now, the right hand side is <math display="block"> \binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k},</math> by Pascal's identity.^[14] On the other hand, if Шаблон:Math, then Шаблон:Math and Шаблон:Math, so we get Шаблон:Math. Thus <math display="block">(x+y)^{n+1} = \sum_{k=0}^{n+1} \binom{n+1}{k} x^{n+1-k} y^k,</math> which is the inductive hypothesis with Шаблон:Math substituted for Шаблон:Mvar and so completes the inductive step.

Generalizations

Newton's generalized binomial theorem

Шаблон:Main Around 1665, Isaac Newton generalized the binomial theorem to allow real exponents other than nonnegative integers. (The same generalization also applies to complex exponents.) In this generalization, the finite sum is replaced by an infinite series. In order to do this, one needs to give meaning to binomial coefficients with an arbitrary upper index, which cannot be done using the usual formula with factorials. However, for an arbitrary number Шаблон:Mvar, one can define <math display="block">{r \choose k}=\frac{r(r-1) \cdots (r-k+1)}{k!} =\frac{(r)_k}{k!},</math> where <math>(\cdot)_k</math> is the Pochhammer symbol, here standing for a falling factorial. This agrees with the usual definitions when Шаблон:Mvar is a nonnegative integer. Then, if Шаблон:Mvar and Шаблон:Mvar are real numbers with Шаблон:Math,^{[Note 1]} and Шаблон:Mvar is any complex number, one has <math display="block">\begin{align}

  (x+y)^r & =\sum_{k=0}^\infty {r \choose k} x^{r-k} y^k \\
  &= x^r + r x^{r-1} y + \frac{r(r-1)}{2!} x^{r-2} y^2 + \frac{r(r-1)(r-2)}{3!} x^{r-3} y^3 + \cdots.
\end{align}</math>

When Шаблон:Mvar is a nonnegative integer, the binomial coefficients for Шаблон:Math are zero, so this equation reduces to the usual binomial theorem, and there are at most Шаблон:Math nonzero terms. For other values of Шаблон:Mvar, the series typically has infinitely many nonzero terms.

For example, Шаблон:Math gives the following series for the square root: <math display="block">\sqrt{1+x} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \frac{5}{128}x^4 + \frac{7}{256}x^5 - \cdots.</math>

Taking Шаблон:Math, the generalized binomial series gives the geometric series formula, valid for Шаблон:Math: <math display="block">(1+x)^{-1} = \frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - x^5 + \cdots.</math>

More generally, with Шаблон:Math, we have for Шаблон:Math:^[15] <math display="block">\frac{1}{(1+x)^s} = \sum_{k=0}^\infty {-s \choose k} x^k = \sum_{k=0}^\infty {s+k-1 \choose k} (-1)^k x^k.</math>

So, for instance, when Шаблон:Math, <math display="block">\frac{1}{\sqrt{1+x}} = 1 - \frac{1}{2}x + \frac{3}{8}x^2 - \frac{5}{16}x^3 + \frac{35}{128}x^4 - \frac{63}{256}x^5 + \cdots.</math>

Replacing Шаблон:Mvar with Шаблон:Mvar yields: <math display="block">\frac{1}{(1-x)^s} = \sum_{k=0}^\infty {s+k-1 \choose k} (-1)^k (-x)^k = \sum_{k=0}^\infty {s+k-1 \choose k} x^k.</math>

So, for instance, when Шаблон:Math, we have for Шаблон:Math: <math display="block">\frac{1}{\sqrt{1-x}} = 1 + \frac{1}{2}x + \frac{3}{8}x^2 + \frac{5}{16}x^3 + \frac{35}{128}x^4 + \frac{63}{256}x^5 + \cdots.</math>

Further generalizations

The generalized binomial theorem can be extended to the case where Шаблон:Mvar and Шаблон:Mvar are complex numbers. For this version, one should again assume Шаблон:Math^{[Note 1]} and define the powers of Шаблон:Math and Шаблон:Mvar using a holomorphic branch of log defined on an open disk of radius Шаблон:Math centered at Шаблон:Mvar. The generalized binomial theorem is valid also for elements Шаблон:Mvar and Шаблон:Mvar of a Banach algebra as long as Шаблон:Math, and Шаблон:Mvar is invertible, and Шаблон:Math.

A version of the binomial theorem is valid for the following Pochhammer symbol-like family of polynomials: for a given real constant Шаблон:Mvar, define <math> x^{(0)} = 1 </math> and <math display="block"> x^{(n)} = \prod_{k=1}^{n}[x+(k-1)c]</math> for <math> n > 0.</math> Then^[16] <math display="block"> (a + b)^{(n)} = \sum_{k=0}^{n}\binom{n}{k}a^{(n-k)}b^{(k)}.</math> The case Шаблон:Math recovers the usual binomial theorem.

More generally, a sequence <math>\{p_n\}_{n=0}^\infty</math> of polynomials is said to be of binomial type if

<math> \deg p_n = n </math> for all <math>n</math>,
<math> p_0(0) = 1 </math>, and
<math> p_n(x+y) = \sum_{k=0}^n \binom{n}{k} p_k(x) p_{n-k}(y) </math> for all <math>x</math>, <math>y</math>, and <math>n</math>.

An operator <math>Q</math> on the space of polynomials is said to be the basis operator of the sequence <math>\{p_n\}_{n=0}^\infty</math> if <math>Qp_0 = 0</math> and <math> Q p_n = n p_{n-1} </math> for all <math> n \geqslant 1 </math>. A sequence <math>\{p_n\}_{n=0}^\infty</math> is binomial if and only if its basis operator is a Delta operator.^[17] Writing <math> E^a </math> for the shift by <math> a </math> operator, the Delta operators corresponding to the above "Pochhammer" families of polynomials are the backward difference <math> I - E^{-c} </math> for <math> c>0 </math>, the ordinary derivative for <math> c=0 </math>, and the forward difference <math> E^{-c} - I </math> for <math> c<0 </math>.

Multinomial theorem

Шаблон:Main The binomial theorem can be generalized to include powers of sums with more than two terms. The general version is

<math display="block">(x_1 + x_2 + \cdots + x_m)^n = \sum_{k_1+k_2+\cdots +k_m = n} \binom{n}{k_1, k_2, \ldots, k_m} x_1^{k_1} x_2^{k_2} \cdots x_m^{k_m}, </math>

where the summation is taken over all sequences of nonnegative integer indices Шаблон:Math through Шаблон:Math such that the sum of all Шаблон:Math is Шаблон:Mvar. (For each term in the expansion, the exponents must add up to Шаблон:Mvar). The coefficients <math> \tbinom{n}{k_1,\cdots,k_m} </math> are known as multinomial coefficients, and can be computed by the formula <math display="block"> \binom{n}{k_1, k_2, \ldots, k_m} = \frac{n!}{k_1! \cdot k_2! \cdots k_m!}.</math>

Combinatorially, the multinomial coefficient <math>\tbinom{n}{k_1,\cdots,k_m}</math> counts the number of different ways to partition an Шаблон:Mvar-element set into disjoint subsets of sizes Шаблон:Math.

Шаблон:Anchor Multi-binomial theorem

When working in more dimensions, it is often useful to deal with products of binomial expressions. By the binomial theorem this is equal to <math display="block"> (x_1+y_1)^{n_1}\dotsm(x_d+y_d)^{n_d} = \sum_{k_1=0}^{n_1}\dotsm\sum_{k_d=0}^{n_d} \binom{n_1}{k_1} x_1^{k_1}y_1^{n_1-k_1} \dotsc \binom{n_d}{k_d} x_d^{k_d}y_d^{n_d-k_d}. </math>

This may be written more concisely, by multi-index notation, as <math display="block"> (x+y)^\alpha = \sum_{\nu \le \alpha} \binom{\alpha}{\nu} x^\nu y^{\alpha - \nu}.</math>

General Leibniz rule

Шаблон:Main

The general Leibniz rule gives the Шаблон:Mvarth derivative of a product of two functions in a form similar to that of the binomial theorem:^[18] <math display="block">(fg)^{(n)}(x) = \sum_{k=0}^n \binom{n}{k} f^{(n-k)}(x) g^{(k)}(x).</math>

Here, the superscript Шаблон:Math indicates the Шаблон:Mvarth derivative of a function, <math>f^{(n)}(x) = \tfrac{d^n}{dx^n}f(x)</math>. If one sets Шаблон:Math and Шаблон:Math, cancelling the common factor of Шаблон:Math from each term gives the ordinary binomial theorem.^[19]

Applications

Multiple-angle identities

For the complex numbers the binomial theorem can be combined with de Moivre's formula to yield multiple-angle formulas for the sine and cosine. According to De Moivre's formula, <math display="block">\cos\left(nx\right)+i\sin\left(nx\right) = \left(\cos x+i\sin x\right)^n.</math>

Using the binomial theorem, the expression on the right can be expanded, and then the real and imaginary parts can be taken to yield formulas for Шаблон:Math and Шаблон:Math. For example, since <math display="block">\left(\cos x + i\sin x\right)^2 = \cos^2 x + 2i \cos x \sin x - \sin^2 x = (\cos^2 x-\sin^2 x) + i(2\cos x\sin x),</math> But De Moivre's formula identifies the left side with <math>(\cos x+i\sin x)^2 = \cos(2x)+i\sin(2x)</math>, so <math display="block">\cos(2x) = \cos^2 x - \sin^2 x \quad\text{and}\quad\sin(2x) = 2 \cos x \sin x,</math> which are the usual double-angle identities. Similarly, since <math display="block">\left(\cos x + i\sin x\right)^3 = \cos^3 x + 3i \cos^2 x \sin x - 3 \cos x \sin^2 x - i \sin^3 x,</math> De Moivre's formula yields <math display="block">\cos(3x) = \cos^3 x - 3 \cos x \sin^2 x \quad\text{and}\quad \sin(3x) = 3\cos^2 x \sin x - \sin^3 x.</math> In general, <math display="block">\cos(nx) = \sum_{k\text{ even}} (-1)^{k/2} {n \choose k}\cos^{n-k} x \sin^k x</math> and <math display="block">\sin(nx) = \sum_{k\text{ odd}} (-1)^{(k-1)/2} {n \choose k}\cos^{n-k} x \sin^k x.</math>There are also similar formulas using Chebyshev polynomials.

Series for e

The [[e (mathematical constant)|number Шаблон:Mvar]] is often defined by the formula <math display="block">e = \lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n.</math>

Applying the binomial theorem to this expression yields the usual infinite series for Шаблон:Mvar. In particular: <math display="block">\left(1 + \frac{1}{n}\right)^n = 1 + {n \choose 1}\frac{1}{n} + {n \choose 2}\frac{1}{n^2} + {n \choose 3}\frac{1}{n^3} + \cdots + {n \choose n}\frac{1}{n^n}.</math>

The Шаблон:Mvarth term of this sum is <math display="block">{n \choose k}\frac{1}{n^k} = \frac{1}{k!}\cdot\frac{n(n-1)(n-2)\cdots (n-k+1)}{n^k}</math>

As Шаблон:Math, the rational expression on the right approaches Шаблон:Math, and therefore <math display="block">\lim_{n\to\infty} {n \choose k}\frac{1}{n^k} = \frac{1}{k!}.</math>

This indicates that Шаблон:Mvar can be written as a series: <math display="block">e=\sum_{k=0}^\infty\frac{1}{k!}=\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots.</math>

Indeed, since each term of the binomial expansion is an increasing function of Шаблон:Mvar, it follows from the monotone convergence theorem for series that the sum of this infinite series is equal to Шаблон:Mvar.

Probability

The binomial theorem is closely related to the probability mass function of the negative binomial distribution. The probability of a (countable) collection of independent Bernoulli trials <math>\{X_t\}_{t\in S}</math> with probability of success <math>p\in [0,1]</math> all not happening is

<math> P\left(\bigcap_{t\in S} X_t^C\right) = (1-p)^{|S|} = \sum_{n=0}^{|S|} {|S| \choose n} (-p)^n.</math>

An upper bound for this quantity is <math> e^{-p|S|}.</math>^[20]

In abstract algebra

The binomial theorem is valid more generally for two elements Шаблон:Math and Шаблон:Math in a ring, or even a semiring, provided that Шаблон:Math. For example, it holds for two Шаблон:Math matrices, provided that those matrices commute; this is useful in computing powers of a matrix.^[21]

The binomial theorem can be stated by saying that the polynomial sequence Шаблон:Math is of binomial type.

In popular culture

The binomial theorem is mentioned in the Major-General's Song in the comic opera The Pirates of Penzance.
Professor Moriarty is described by Sherlock Holmes as having written a treatise on the binomial theorem.
The Portuguese poet Fernando Pessoa, using the heteronym Álvaro de Campos, wrote that "Newton's Binomial is as beautiful as the Venus de Milo. The truth is that few people notice it."^[22]
In the 2014 film The Imitation Game, Alan Turing makes reference to Isaac Newton's work on the binomial theorem during his first meeting with Commander Denniston at Bletchley Park.

Notes

Шаблон:Reflist

References