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Can anyone think of an easy way to find the values of $x$ for which the following series converges?

$$\sum_{n=1}^\infty (3x+1)^{-3n}$$

I'm thinking some convergence test (root test, perhaps?) but I'm a little rusty on these things and I'm not sure how to approach it.

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Take a look at this: – Jack Apr 16 '12 at 4:17
up vote 1 down vote accepted

Note that $\sum\limits_{n=1}^\infty (3x+1)^{-3n}=\sum\limits_{n=1}^\infty y^n$ where $y=(3x+1)^{-3}$, which is a standard geometric series.

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Write $(3x+1)^{-3n}=\bigl((3x+1)^{-3}\bigr)^n$, set $r=(3x+1)^{-3}$, and try to recognize the series as a well-known type.

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