Superbird
Fire emblem is great
For example, ∑((n * x^n) / n^2), n=1 to infinity. I need to find the radius of convergence for x, and the bounds of that convergence.
I can use ratio test to figure out the answer to this problem:
= ((n+1) * x^(n+1) * n^2) / ((n+1)^2 * x^n * n)
= x * n/(n+1).
—> |x * n/n+1| < 1
—> -1 < x < 1
Thus the radius of convergence is 1, and the bounds are (-1,1).
...or are they. This is what I have trouble with -- do I use square brackets or round brackets, and how do I figure out which for each bound? I know it has something to do with figuring out whether the series converges for each bound of x, but does that necessarily mean I have to go through the trouble of figuring that out? For instance, would I now have to evaluate
∑((n * (-1)^n) / n^2) and ∑((n * 1^n) / n^2)
again?
(ok, I know it's a bad example but it's an example of the problem I want to solve.) I know that the first of those, with x = -1, is convergent because of the alternating series test, whereas the second, with x = 1, is divergent because p-series. So the bounds would be [-1,1). But do I have to go through that much effort with every single problem, or is there an easier way?
I can use ratio test to figure out the answer to this problem:
= ((n+1) * x^(n+1) * n^2) / ((n+1)^2 * x^n * n)
= x * n/(n+1).
—> |x * n/n+1| < 1
—> -1 < x < 1
Thus the radius of convergence is 1, and the bounds are (-1,1).
...or are they. This is what I have trouble with -- do I use square brackets or round brackets, and how do I figure out which for each bound? I know it has something to do with figuring out whether the series converges for each bound of x, but does that necessarily mean I have to go through the trouble of figuring that out? For instance, would I now have to evaluate
∑((n * (-1)^n) / n^2) and ∑((n * 1^n) / n^2)
again?
(ok, I know it's a bad example but it's an example of the problem I want to solve.) I know that the first of those, with x = -1, is convergent because of the alternating series test, whereas the second, with x = 1, is divergent because p-series. So the bounds would be [-1,1). But do I have to go through that much effort with every single problem, or is there an easier way?
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