Power Series and Exact Values of Numerical Series

Key Questions

How do you use a power series to find the exact value of the sum of the series $π - \frac{π^{2}}{2} + \frac{π^{4}}{4!} - \frac{π^{6}}{6!} + \dots$ $pi-pi^2/2+pi^4/(4!) -pi^6/(6!) + …$ ?

Since

$1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \dots = cos x$ $1-x^2/{2!}+{x^4}/{4!}-{x^6}/{6!}+cdots=cosx$ ,

$1 - \frac{π^{2}}{2!} + \frac{π^{4}}{4!} - \frac{π^{6}}{6!} + \dots = cos (π) = - 1$ $1-pi^2/{2!}+{pi^4}/{4!}-{pi^6}/{6!}+cdots=cos(pi)=-1$

Therefore,

$π - \frac{π^{2}}{2!} + \frac{π^{4}}{4!} - \frac{π^{6}}{6!} + \dots$ $pi-pi^2/{2!}+{pi^4}/{4!}-{pi^6}/{6!}+cdots$

$= π - 1 + (1 - \frac{π^{2}}{2!} + \frac{π^{4}}{4!} - \frac{π^{6}}{6!} + \dots)$ $=pi-1+(1-pi^2/{2!}+{pi^4}/{4!}-{pi^6}/{6!}+cdots)$

$= π - 1 + (- 1) = π - 2$ $=pi-1+(-1)=pi-2$

I hope that this was helpful.

Wataru · · Sep 28 2014
How do you use a power series to find the exact value of the sum of the series $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$ $1-1/2+1/3-1/4 + …$ ?

Alternating Harmonic Series

$\infty \sum n = 1 \frac{{(- 1)}^{n - 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots = ln 2$ $sum_{n=1}^infty(-1)^{n-1}/n=1-1/2+1/3-1/4+cdots=ln2$

Since

$ln (1 - x) = - \infty \sum n = 1 \frac{x^{n}}{n}$ $ln(1-x)=-sum_{n=1}^infty{x^n}/n$ ,

by setting $x = - 1$ $x=-1$ ,

$ln 2 = - \infty \sum n = 1 \frac{{(- 1)}^{n}}{n} = \infty \sum n = 1 \frac{{(- 1)}^{n - 1}}{n}$ $ln2=-sum_{n=1}^infty{(-1)^n}/n=sum_{n=1}^infty(-1)^{n-1}/n$

Wataru · · Sep 25 2014
How do you use a power series to find the exact value of the sum of the series $1 + 2 + \frac{4}{2!} + \frac{8}{3!} + \frac{16}{4!} + \dots$ $1+2+4/(2!) +8/(3!) +16/(4!) + …$ ?

Since

$1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \dots = e^{x}$ $1+x+x^2/{2!}+x^3/{3!}+x^4/{4!}+cdots=e^x$ ,

by replacing $x$ $x$ by $2$ $2$ ,

$1 + 2 + \frac{2^{2}}{2!} + \frac{2^{3}}{3!} + \frac{2^{4}}{4!} + \dots = e^{2}$ $1+2+2^2/{2!}+2^3/{3!}+2^4/{4!}+cdots=e^2$ .

I hope that this was helpful.

Wataru · 1 · Oct 10 2014

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Power Series and Exact Values of Numerical Series

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Power Series