- #1
vande060
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Homework Statement
Ive been able to do every single problem in my homework to the point where I have to test the edges of the interval of convergence. I have not been able to figure out a single one of the problems at the point of testing the edge of convergence, and I am to the point of pulling my hair out, so I really need an explanation of what to do at that point. Ill give an example of a problem.find the power series for representation for the function
f(x) = x/(9+x^2)
Homework Equations
The Attempt at a Solution
f(x) = x/(9+x2)
f(x) = x/(9 -(-x2)
f(x) = x/9(1-(-x2/9)
(∞, n=0) (x/9)∑ (-1)n [(x2)/(9)] n
(∞, n=0) ∑ (-1)n (x2n+1)/ (9n+1)
then i do a quicktest to find the radius of convergence
lim n--> ∞ |cn/cn+1|
lim n--> ∞ |[(-1)n/(9n+1)] * [(9n+2)/(-1)n+1] = 9
|x2| < |9|
-3 <x < 3
this is where i start to feel bad, i know the radius of convergence is 3 and the interval of convergence is (-3,3), but I have to test the edges of convergence, and I can't seem to do it. I know i should plug -3 and 3 back into the power series and test for convergence, but i have no idea how to test these two series for convergence
(∞, n=0) ∑ (-1)n ((-3)2n+1)/ (9n+1)
(∞, n=0) ∑ (-1)n ((3)2n+1)/ (9n+1)