Binomial theorem

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In mathematics, the binomial theorem is a formula that provides the development of the $n$ - That's the power of a binomial. ${displaystyle nin mathbb {Z} ^{+}}$ . According to the theorem, it is possible to expand the power ${displaystyle (x+y)^{n}}$ in a sum that implies terms of form ${displaystyle ax^{b}y^{c}}$ where the exponents ${displaystyle b,cin mathbb {N} }$ I mean, they're natural numbers with ${displaystyle b+c=n}$ and the coefficient $a$ of each term is a positive integer that depends on $n$ and $b$ . When an exponent is zero, the corresponding power is usually omitted from the term.

The coefficient $a$ in the terms of ${displaystyle x^{b}y^{c}-x^{c}y^{b}}$ is known as the binomial coefficient ${textstyle {binom {n}{b}}}$ or ${textstyle {binom {n}{c}}}$ (the two have the same value).

Theorem

This theorem establishes that any power of a binomial ${displaystyle x+y}$ can be expanded in a sum of form:

{displaystyle {begin{aligned}(x+y)^{n}&=sum _{k=0}^{n}{n choose k}x^{n-k}y^{k}\&={n choose 0}x^{n}+{n choose 1}x^{n-1}y+{n choose 2}x^{n-2}y^{2}+cdots +{n choose n-1}xy^{n-1}+{n choose n}y^{n}end{aligned}}}

where

{displaystyle {binom {n}{k}}}

is the binomial coefficient, which represents the number of ways to choose $k$ elements of a set with $n$ elements.

Using the formula for calculating said coefficient, the following equation is obtained:

{displaystyle (x+y)^{n}=sum _{k=0}^{n}{frac {n!}{k!(n-k)!}}x^{n-k}y^{k}}

To obtain the expansion of the powers of a subtraction, just take ${displaystyle -y}$ instead of $y$ in terms with odd powers:

{displaystyle (x-y)^{n}=displaystyle sum _{k=0}^{n}(-1)^{k}{frac {n!}{k!(n-k)!}}x^{n-k}y^{k}}

Examples

As examples, for ${displaystyle n=2,3,4,5}$ , using the coefficients of the triangle of Pascal these results are obtained:

{displaystyle {begin{aligned}(x+y)^{2}&=x^{2}+2xy+y^{2}\(x+y)^{3}&=x^{3}+3x^{2}y+3xy^{2}+y^{3}\(x+y)^{4}&=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}\(x+y)^{5}&=x^{5}+5x^{4}y+10x^{3}y^{2}+10x^{2}y^{3}+5xy^{4}+y^{5}end{aligned}}}

Generalized Binomial Theorem (Newton)

Isaac Newton generalized the formula for real exponents, considering an infinite series:

{displaystyle {(x+y)^{r}=sum _{k=0}^{infty }{r choose k}x^{r-k}y^{k}}}

where $r$ can be any real number, not necessarily positive or integer, and the coefficients are given by the product:

{displaystyle {r choose k}={1 over k!}prod _{n=0}^{k-1}(r-n)={frac {r(r-1)(r-2)cdots (r-k+1)}{k!}}={frac {r!}{(r-k)!k!}}}

The expansion for reciprocal power is as follows:

{displaystyle {frac {1}{(1-x)^{r}}}=sum _{k=0}^{infty }{r+k-1 choose k}x^{k}}

The amount in (3converge and equality is true whenever real or complex numbers $x$ e $y$ are close enough, in the sense that the absolute value of ${displaystyle {frac {x}{y}}}$ be less than one.

Multinomial Theorem

Main article: Multinomial Theorem

The binomial theorem can be generalized to include powers of sums of more than two terms. In general:

{displaystyle (x_{1}+x_{2}+cdots +x_{m})^{n}=sum _{k_{1}+k_{2}+cdots +k_{m}=n}{n choose k_{1},k_{2},ldotsk_{m}}x_{1}^{k_{1}}x_{2}^{k_{2}}cdots x_{m}^{k_{m}}.}

In this formula, the sum is taken on all natural integer values from $k_{1}$ until ${displaystyle k_{m}}$ such that the sum of all these values is equal to $n$ . Summary coefficients, known as multi-year ratios are calculated according to the formula:

{displaystyle {n choose k_{1},k_{2},ldotsk_{m}}={frac {n!}{k_{1}!cdot k_{2}!cdots k_{m}!}}.}

From the point of view of the combination, the multinomial coefficient counts the number of different ways to divide a set of $n$ elements in subsets set of sizes ${displaystyle k_{1},k_{2},ldotsk_{m}}$

Multi-binomial theorem

It is often useful, when working in more than one dimension, to use products of binomial expressions. By the binomial theorem, this is equal to:

{displaystyle (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}}.}

The above formula can be written using the multi-index notation as follows:

{displaystyle (x+y)^{alpha }=sum _{nu leq alpha }{binom {alpha }{nu }}x^{nu }y^{alpha -nu }.}

Generalized product rule (or Leibniz formula of the nth derivative)

Main article: Rule of Leibniz

The Leibniz General Rule provides the $n-$ the product of two functions $f(x)$ and $g(x)$ similar to the theorem of the binomial:

{displaystyle (fcdot g)^{(n)}(x)=sum _{k=0}^{n}{binom {n}{k}}f^{(n-k)}(x)g^{(k)}(x).}

In this equality, the superscript $n$ indicates $n-$ esima derived from a function. If we do ${displaystyle f(x)=e^{ax}}$ and ${displaystyle f(x)=e^{bx}}$ the common factor is cancelled on both sides of equality ${displaystyle e^{(a+b)x}}$ and you get the theorem from the binomial.

Applications

Multiple angle identities

For complex numbers the binomial theorem can be combined with De Moivre's Formula to provide multiple angle identities for the sine and cosine functions. According to De Moivre's formula:

{displaystyle cos left(nxright)+ioperatorname {sen} left(nxright)=left(cos x+ioperatorname {sen} xright)^{n}.}

Using the binomial theorem, the expression on the right hand side can be expanded and then the real and imaginary parts are extracted to obtain the multiple angle formulas. Given that:

{displaystyle left(cos x+ioperatorname {sen} xright)^{2}=cos ^{2}(x)+2icos(x)operatorname {sen}(x)-operatorname {sen} ^{2}(x)}

Comparing this equality with De Moivre's formula, it becomes clear that:

{displaystyle {begin{aligned}cos(2x)&=cos ^{2}(x)-operatorname {sen} ^{2}(x)\operatorname {sen}(2x)&=2cos(x)operatorname {sen}(x)end{aligned}}}

which are the usual double angle identities.

Similarly:

{displaystyle left(cos x+ioperatorname {sen} xright)^{3}=cos ^{3}(x)+3icos ^{2}(x)operatorname {sen}(x)-3cos(x)operatorname {sen} ^{2}(x)-ioperatorname {sen} ^{3}(x)}

Comparing with the statement of De Moivre's formula, by separating the real and imaginary parts of the result:

{displaystyle {begin{aligned}cos(3x)&=cos ^{3}(x)-3cos(x)operatorname {sen} ^{2}(x)\operatorname {sen}(3x)&=3cos ^{2}(x)operatorname {sen}(x)-operatorname {sen} ^{3}(x)end{aligned}}}

In general,

{displaystyle cos(nx)=sum _{k{text{ par}}}(-1)^{k/2}{n choose k}cos ^{n-k}(x)operatorname {sen} ^{k}(x)}

and

{displaystyle operatorname {sen}(nx)=sum _{k{text{ impar}}}(-1)^{(k-1)/2}{n choose k}cos ^{n-k}(x)operatorname {sen} ^{k}(x)}

Series for e

The number e is usually defined by the equation:

{displaystyle e=lim _{nto infty }left(1+{frac {1}{n}}right)^{n}}

Applying the theorem of the binomial to this expression we get the infinite series $e$ . In particular:

{displaystyle 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}}}.}

The k-th term of this sum is:

{displaystyle {begin{aligned}{n choose k}{frac {1}{n^{k}}}&={frac {n!}{k!cdot (n-k)!}}cdot {frac {1}{n^{k}}}\&={frac {1}{k!}}cdot {frac {n(n-1)(n-2)cdots (n-k+1)cdot (n-k)!}{(n-k)!}}cdot {frac {1}{n^{k}}}\&={frac {1}{k!}}cdot {frac {n(n-1)(n-2)cdots (n-k+1)}{n^{k}}}end{aligned}}}

Like the number $n$ tends to infinite ( ${displaystyle nto infty }$ ), the rational expression to the right is approx.1

{displaystyle {begin{aligned}lim _{nto infty }{frac {n(n-1)(n-2)cdots (n-k+1)}{n^{k}}}&=lim _{nto infty }{frac {n^{k}+a_{1}cdot n^{k-1}+a_{2}cdot n^{k-2}cdots a_{k+2}cdot n^{1}+a_{k+1}cdot n^{0}}{n^{k}}}\&=lim _{nto infty }{frac {n^{k}}{n^{k}}}+lim _{nto infty }{frac {a_{1}cdot n^{k-1}}{n^{k}}}cdots lim _{nto infty }{frac {a_{k+1}cdot n^{0}}{n^{k}}}\&=1+0cdots 0\&=1end{aligned}}}

And therefore, when n tends to infinity, each k-th term reduces to:

{displaystyle lim _{nto infty }{n choose k}{frac {1}{n^{k}}}={frac {1}{k!}}.}

Which indicates that $e$ you can write as an infinite series:

{displaystyle e=sum _{k=0}^{infty }{frac {1}{k!}}={frac {1}{0!}}+{frac {1}{1!}}+{frac {1}{2!}}+{frac {1}{3!}}+cdots.}

Probability

The binomial theorem is closely related to the probability function of mass of negative binomial distribution. The probability of an independent Bernoulli test collection (counter) ${displaystyle {X_{t}}_{tin S}}$ with probability of success ${displaystyle pin [0,1]}$ Don't happen. ${displaystyle Pleft(bigcap _{tin S}X_{t}^{C}right)=(1-p)^{|S|}=sum _{n=0}^{|S|}{|S| choose n}(-p)^{n}}$

A useful upper limit for this amount is ${displaystyle e^{pn}}$ .

History

Attributed to Isaac Newton, the theorem was actually first discovered by Al-Karjí around the year 1000. Applying John Wallis's methods of interpolation and extrapolation to new problems, Newton used the concepts of generalized exponents by which a polynomial expression was transformed into an infinite series. Thus he was in a position to demonstrate that a large number of already existing series were particular cases, either by differentiation or by integration.

In the winter of 1664 and 1665, Newton who was in his home in Lincolnshire extended the binomial expansion in the case that $n$ is a rational number and in the next autumn, when the exponent is a negative number. For both cases, it was found that the resulting expression was a series of infinite terms.

In the case of negative exponents, Newton used the stepped form of Pascal's Triangle, which was exposed by the German mathematician Michael Stifel in his work Arithmetica Integra:

${displaystyle {begin{matrix}n=0:&1&0&0&0&0&0&0&0&cdots \n=1:&1&1&0&0&0&0&0&0&cdots \n=2:&1&2&1&0&0&0&0&0&cdots \n=3:&1&3&3&1&0&0&0&0&cdots \n=4:&1&4&6&4&1&0&0&0&cdots \\end{matrix}}}$

In this form it is easy to see that the sum of the j-th element and the (j-1)-th element of a row results in the j-th element of the row below. Newton extended this table upward, finding the difference between the j-th element in a row and the (j-1)-th element in the row above the previous one, placing the result as the j-th element in that row above. So, he was able to get this new table:

${displaystyle {begin{array}{rrrrrrrrr}n=-4:&1&-4&10&-20&35&-56&84&cdots \n=-3:&1&-3&6&-10&15&-21&28&cdots \n=-2:&1&-2&3&-4&5&-6&7&cdots \n=-1:&1&-1&1&-1&1&-1&1&cdots \n=0:&1&0&0&0&0&0&0&cdots \n=1:&1&1&0&0&0&0&0&cdots \n=2:&1&2&1&0&0&0&0&cdots \n=3:&1&3&3&1&0&0&0&cdots \n=4:&1&4&6&4&1&0&0&cdots \end{array}}}$

Realizing that the number series had no end, Newton concluded that for a negative integer the series is infinite what indicates, in fact, that if the sum ${displaystyle (x+y)}$ represented the binomial ${displaystyle (1+x)}$ the result obtained is valid if $x$ is between -1 and 1. Yeah. $n$ is a rational number, studying the obtained pattern, Newton was able to obtain binomial coefficients for fractions such as ${frac {1}{2}}$ , ${displaystyle {frac {3}{2}}}$ and ${displaystyle {frac {5}{2}}}$ For example. In that case, if ${displaystyle n={frac {1}{2}}}$ , coefficients are $1$ , ${displaystyle -{frac {1}{2}}}$ , ${displaystyle {frac {1}{8}}}$ , ${displaystyle -{frac {5}{128}}}$ etc. Newton was able to check that if he multiplied the expansion ${frac {1}{2}}$ , by itself, obtained precisely the case in which ${displaystyle n=1}$ .

From this discovery, Newton had the intuition that it could be operated with infinite series in the same way as with fine polynomial expressions. Newton never published this theorem. Wallis did it for the first time in 1685 in his Algebra, accusing Newton of this discovery. The binary theorem for ${displaystyle n=2}$ is located in the Elements of Euclides (300 BC) and the term "binomial coefficient" was introduced by Stifel.

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