Infinite Series

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September 9th 2010, 02:30 AM
heatly

Infinite Series

The Problem:
summation as 'n' goes from 2 to infinity of n^20/2^n .

The ans says it converges to 1/2 but cant see how it does this since the terms seem to be increasing in magnitude when you plug numbers in.
September 9th 2010, 02:38 AM
yeKciM

Quote:

Originally Posted by heatly

The Problem:
summation as 'n' goes from 2 to infinity of n^20/2^n .

The ans says it converges to 1/2 but cant see how it does this since the terms seem to be increasing in magnitude when you plug numbers in.

$\displaystyle \sum _{n=2} ^{\infty} \frac {n^{20}}{2^n}$

so you have that your a_n is

$\displaystyle a_n = \frac {n^{20}}{2^n}$

now apply some of the tests for convergence ...

you can use Cauchy's test of convergence :

$\displaystyle q= \lim _{n\to \infty} \sqrt [n] {a_n} = \lim_{n \to \infty }\sqrt [n] { \frac {n^{20}}{2^n}} = \left\{\begin{matrix} q<1 & converges \\ q>1 & diverges \\ q=1 & ??? \end{matrix}\right.$

or with D'Alambert's test for convergence :

$\displaystyle q= \lim_{n \to \infty } \frac {a_{n+1}}{a_n} =\lim_{n \to \infty } \frac {\frac {(n+1)^{20}}{2^{n+1}} }{\frac {n^{20}}{2^n}} = \left\{\begin{matrix} q<1 & converges \\ q>1 & diverges \\ q=1 & ??? \end{matrix}\right.$

or with Raabe's test of convergence :
$\displaystyle q= \lim_{n \to \infty } n(\frac {a_{n}}{a_{n+1} }-1 ) =\lim_{n \to \infty } n ( \frac {\frac {n^{20}}{2^n} }{\frac {(n+1)^{20}}{2^{n+1}}}-1 ) = \left\{\begin{matrix} q>1 & converges \\ q<1 & diverges \\ q=1 & ??? \end{matrix}\right.$

or you can show by the definition that sum converges or diverges :D (with partial sums .... )
September 9th 2010, 03:30 AM
mr fantastic

Quote:

Originally Posted by heatly

The Problem:
summation as 'n' goes from 2 to infinity of n^20/2^n .

The ans says it converges to 1/2 but cant see how it does this since the terms seem to be increasing in magnitude when you plug numbers in.

The first term of the series is (2^20)/4 = 2^18. So the claim that the series converges to a value of 1/2 is absolute rubbish.
September 9th 2010, 03:52 AM
yeKciM

Quote:

Originally Posted by mr fantastic

The first term of the series is (2^20)/4 = 2^18. So the claim that the series converges to a value of 1/2 is absolute rubbish.

$\displaystyle \sum _{n=2} ^{\infty} \frac {n^{20}}{2^n}$

so based on D'Alambert test of convergence :

$\displaystyle q= \lim_{n \to \infty } \frac {a_{n+1}}{a_n} =\lim_{n \to \infty } \frac {\frac {(n+1)^{20}}{2^{n+1}} }{\frac {n^{20}}{2^n}} =\lim_{n \to \infty } \frac {\frac {(n+1)^{20}}{2^n\cdot 2}} {\frac {n^{20}}{2^n}} = \lim_{n \to \infty } \frac {n^{20 } + ... } { 2n^{20} } = \frac {1}{2}$

now because $q = \frac {1}{2} <1$ based on D'Alambert test of convergence sum $\displaystyle \sum _{n=2} ^{\infty} \frac {n^{20}}{2^n}$ converges :D:D:D

maybe that's what ans was about :D