Using the series comparison test on $\sum_1^n\frac{2}{3+5n}$

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!

• Wait, so you are looking to see if the series converges or diverges? – mattos Jul 29 '15 at 4:38
• yes, to determine if the series converges or diverges – junhaotee Jul 29 '15 at 4:46
• 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.) – Mirko Jul 29 '15 at 4:48
• modified, thanks – junhaotee Jul 29 '15 at 4:56

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:

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

Hint

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

I am sure that you can take from here.