[SOLVED] Power Series(Radius and interval of convergence)

**nameck** · Dec 9th 2009, 06:23 PM

hey there..
i've try this question already...
i got radius of convergence, R = 1/8

at x = 1/8,
converges...

at x = -1/8
i don't how to do this one..
please help me..

**tonio** · Dec 9th 2009, 06:26 PM

Originally Posted by nameck

hey there..
i've try this question already...
i got radius of convergence, R = 1/8

at x = 1/8,
converges...

at x = -1/8
i don't how to do this one..
please help me..

At $x=-\frac{1}{8}$ you get $\sum\limits_{n=1}^\infty\frac{(-1)^n}{n!} =$ a Leibnitz series and thus convergent.

Tonio

**nameck** · Dec 9th 2009, 06:33 PM

the question is..
find the radius of convergence and interval of convergence of the
power series (attachment)

tonio.. is (8^n) / (-8) = -1^n?
is my radius of interval is correct?

**VonNemo19** · Dec 9th 2009, 06:36 PM

$a_n=\frac{8^n}{n!}x^n$

$|\frac{a_{n+1}}{a_n}|=|\frac{8^{n+1}}{(n+1)!}x^{n+ 1}\cdot\frac{n!}{8^nx^n}|=\frac{8|x|}{n+1}$

Now, $\lim_{n\to\infty}\frac{8|x|}{n+1}=0<1$ $\forall{x}\in(-\infty,\infty)$ .

Therefore, the series converges $\forall{x}$ by the ratio test.

This implies an infinte radius.

**nameck** · Dec 9th 2009, 06:45 PM

how did u get radius from infity to -infinity?
i haven't learn this kind solution(ration test to find radius and interval)

**VonNemo19** · Dec 9th 2009, 07:38 PM

Originally Posted by nameck

how did u get radius from infity to -infinity?
i haven't learn this kind solution(ration test to find radius and interval)

Well, what tests for convergence do you know?

**tonio** · Dec 10th 2009, 02:01 AM

Originally Posted by nameck

the question is..
find the radius of convergence and interval of convergence of the
power series (attachment)

tonio.. is (8^n) / (-8) = -1^n?

No, and this is not what you have. At $x=-\frac{1}{8}$ you have $\frac{8^n}{n!}\left(-\frac{1}{8}\right)^n=\left(\frac{8}{-8}\right)^n\frac{1}{n!}=\frac{(-1)^n}{n!}$

is my radius of interval is correct?

No. The convergence radius is what VonNemo wrote and for the reasons he/she wrote.

Tonio

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