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Here is the given series 3/(9n+1), decide whether it converges or diverges. I used the ratio test only to end up with the ratio=1. I know this is harmonic series but it is smaller than 1/n, therefore i cannot conclude it diverges. Please help!!

4 Answers 4

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Hint

$\dfrac{3}{9 n+1} \geqslant \dfrac{3}{12n}=\dfrac{1}{4}\cdot \dfrac{1}{n}$

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$ \frac{3}{9n+1} \ge \frac{1}{9n+1} \ge \frac{1}{9n+9} \ge \frac{1}{9(n+1)} \ge \frac{1}{9} \frac{1}{n+1} $

At which point you should be able to figure that out...

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    are u trying to $s$how that this series is greater than the divergent series therefore it is divergent?2012-10-28
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We have $ \frac{3}{9n+1}\geq \frac{3}{9n+9}=\frac{1}{3(n+1)}. $ Since $\displaystyle\sum_{n=1}^{\infty}\frac{1}{3(n+1)}$ is disvergent, $\displaystyle\sum_{n=1}^{\infty}\frac{3}{9n+1}$.

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    @d13: You are welcome.2012-10-28
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The ratio and root tests are very crude and won't work here. Have you tried comparison oar limit comparison?

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    ahh i see, thanks a lot!!2012-10-29