power series of rep

Apr 2010
21
0
Sum (-1)^n/(n 2^n) x^n

0. Find the interval of convergence of the power series defining f(x).

0. Find a power seies for f'(x) and determine its interval of convergence.

0. Find a power seies for integral 0 to x, f(t) dt and determine its interval of convergence.

any help would be appreciated!
 

matheagle

MHF Hall of Honor
Feb 2009
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The ratio test gives you \(\displaystyle -1<x/2<1\)

giving you (-2,2), now check the end points.
 
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Apr 2009
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Boston
Sum (-1)^n/(n 2^n) x^n

0. Find the interval of convergence of the power series defining f(x).

0. Find a power seies for f'(x) and determine its interval of convergence.

0. Find a power seies for integral 0 to x, f(t) dt and determine its interval of convergence.

any help would be appreciated!

Set up the Ratio Test.

\(\displaystyle |\frac{a_{n+1}}{a_n}|=|x| \frac{n}{2(n+1)}\)

which approaches

\(\displaystyle \frac{|x|}{2}\) as n tends to infinity

Set this less than 1, then determine the radius and interval of convergence.

As to parts b) and c), there is a theorem that says that the radius of convergence remains the same for both the derivative and the antiderivative.

Good luck!
 
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Reactions: twofortwo
Apr 2010
21
0
thank you. but how do i find the power series for the integral and derivative? i can use the theorem afterwards for the intervals but how do i go about finding the power series of derivitive and integral?
 

matheagle

MHF Hall of Honor
Feb 2009
2,763
1,146
Set up the Ratio Test.

\(\displaystyle |\frac{a_{n+1}}{a_n}|=|x| \frac{n}{2(n+1)}\)

which approaches

\(\displaystyle \frac{|x|}{2}\) as n tends to infinity

Set this less than 1, then determine the radius and interval of convergence.

As to parts b) and c), there is a theorem that says that the radius of convergence remains the same for both the derivative and the antiderivative.

Good luck!
The interval remains the same, but you can gain an endpoint or two when
you integrate. Likewise you can lose endpoint when you differentiate.
So you need to check the endpoints after integrating and differentiating.