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Summation problems

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1.

Determine all values of a for which $$\displaystyle\sum_{n=1}^{\infty}\left(\dfrac{2}{a}\right)^n$$ converges. I believe this to be (2,inf)
2.

Determine all values of a for which $$\displaystyle\sum_{n=1}^{\infty}\left(\dfrac{2}{n}\right)^a$$  converges. I believe this to be (0,inf)

Mar 31, 2021

#1
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1. Almost right; you forgot negative values

(just as a reminder, the sum $$\displaystyle\sum_{n=1}^{\infty}x^n$$ where x is a constant only converges when $$-1 by the ratio test.) So the answer would be \((-\infty, -2)\bigcup(2, \infty)$$

2. The sum $$\displaystyle\sum_{n=1}^{\infty}\left(\dfrac{y}{n}\right)^a$$, where y is a constant, only converges when $$a>1$$, so the answer is $$(1, \infty)$$

Mar 31, 2021