`sum_(n=1)^oo (3/p)^n` Find the positive values of p for which the series converges.

Expert Answers

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This series is the sum of an infinite geometrical progression with the common ratio of  `3/p.` It is well known that such a series converges if and only if its common ratio is less than `1` by the absolute value.

In this problem we have the condition  `|3/p| lt 1,` or  `|p| gt 3.` Because we are asked about positive p's, we have  `p gt 3.`

The answer: for positive `p` this series converges if and only if  `p gt 3.` 

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