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Interval of Convergence Q: Find the interval of convergence of this power series: 2x +...
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Honors, Dean's List
Q: Find the interval of convergence of this power series:
2x + `x^(2)` + `(8)/(9)` `x^(3)` + `x^(4)` + ... = `sum_(n=1)^oo` `(2^n)/(n^2)` `x^(n)`
Posted by Rocky52 on May 11, 2013 at 6:09 AM (Answer #1)
High School Teacher
This is power series around 0 so it will converge for `x=0`. Now we need to find the radius of convergence `r` which tells us how far away from 0 will the series converge. There are several different ways to find convergence radius and I will show you one of them.
In your case `a_n=2^n/n^2` so we have
` `` ` So radius of convergence is `r=1/2` which means that your series will converge for `|x-0|<1/2=>|x|<1/2`.
The series is convergent for `|x|<1/2` (`-1/2<x<1/2` )
Posted by tiburtius on May 11, 2013 at 8:01 AM (Answer #2)
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