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Is my answer correct?

Q:Determine whether $\sum_1^n\frac{2}{3+5n}$ converges or diverges.

A:$\frac{2}{6n}$ < $\frac{2}{3+5n}$,

since $\sum_1^n\frac{2}{6n}$ is a harmonic series and hence diverges, then $\sum_1^n\frac{2}{3+5n}$ will diverge also.

It can be done using integral test but I'm trying to do it using comparison test, thanks!

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    $\begingroup$ Wait, so you are looking to see if the series converges or diverges? $\endgroup$ – mattos Jul 29 '15 at 4:38
  • $\begingroup$ yes, to determine if the series converges or diverges $\endgroup$ – junhaotee Jul 29 '15 at 4:46
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    $\begingroup$ what series ... I only see a single fraction? (Except for the answer by @Real who used the $\sum$ notation, but it looks like he introduced it as a guess to what you re asking, you should be clearer.) $\endgroup$ – Mirko Jul 29 '15 at 4:48
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    $\begingroup$ modified, thanks $\endgroup$ – junhaotee Jul 29 '15 at 4:56
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Your inequality is not true. Take $n=1$, it is not true that: $$\frac{2}{6}=\frac{1}{3}<\frac{2}{8}=\frac{1}{4}$$

You could work on writing things more clearly. You want to determine whether the series $\displaystyle \sum^\infty_{n=1} \frac{2}{3+5n}$ diverges or converges.

Suppose we have two series: $\sum(a_n), \sum(b_n)$.

The comparison test states that:

  1. If $\sum(b_n)$ converges and $a_n\leq b_n$ for all $n$, then $\sum(a_n)$ also converges.
  2. If $\sum(b_n)$ diverges and $b_n\leq a_n$ for all $n$, then $\sum(a_n)$ also diverges.

You can use this inequality as a hint:

$$\frac{2}{5n+5n}<\frac{2}{3+5n}$$

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Hint

For $n \geq 1$, you have $$5n <3+5n<8n$$

I am sure that you can take from here.

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