<s>[INST] divergence/convergence $\sum_{k=1}^{\infty} \frac{1}{5k+2}$ 
$$\sum_{k=1}^{\infty} \frac{1}{5k+2}$$

Can I say that $$\sum_{k=1}^{\infty} \frac{1}{5k+2}=\sum_{k=1}\frac{1}{5k}+\sum_{k=1}^{\infty} \frac{1}{2}=\frac{1}{5}\sum_{k=1}^{\infty}\frac{1}{k}+\frac{1}{2}$$
and we know that $\\sum_{k=1}^{\infty} \frac{1}{k}$ diverges so the expression diverges too?
Or should I use the integral test $$\int_{1}^{\infty} \frac{1}{5k+2}=\frac{ln|5k+2|}{5}$$ from 1 to $\infty$=  $\infty$
 [/INST] I would say that 
$$ \sum_{k=1}^\infty\frac{1}{10k}<\sum_{k=1}^\infty \frac{1}{5k+2}$$
which is true because 
$$ \frac{1}{10k}<\frac{1}{5k+2}\:\:\forall k\in \mathbb{N}.$$
$$ \sum_{k=1}^\infty \frac{1}{10k}=\frac{1}{10}\sum_{k=1}^\infty \frac{1}{k}$$
which implies that it diverges. So, because 
$$ \sum_{k=1}^\infty\frac{1}{5k+2}$$
is greater term by term than a series which diverges, it too diverges.
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