$$\frac{2}{1^p}\ + \frac{3}{2^p}\ + \frac{4}{3^p}\ +\ldots\,.$$
I can see that the $nth$ term is $\frac{n+1}{n^p}$
How do I test for its convergence?
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JimmyK4542
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Diya
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Should the third term be $\frac{4}{3^p}$? – JimmyK4542 Aug 27 '14 at 03:42
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Check the integral test. – Mhenni Benghorbal Aug 27 '14 at 03:47
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Thanks but the intergral test is not included within my syllabus I'm afraid. I used the p-series test. :) – Diya Aug 27 '14 at 03:50
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Hint: $\displaystyle\sum_{n=1}^{\infty}\dfrac{n+1}{n^p} = \sum_{n=1}^{\infty}\left[\dfrac{n}{n^p}+\dfrac{1}{n^p}\right] = \sum_{n=1}^{\infty}\left[\dfrac{1}{n^{p-1}}+\dfrac{1}{n^p}\right]$. Now apply the $p$-test.
JimmyK4542
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Convergence will depend on the value of $p$. So your answer will be something like "The series converges for [conditions on $p$] and diverges for [opposite conditions on $p$]". – JimmyK4542 Aug 27 '14 at 03:46
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