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how would I be able to prove that, using the comparison test,

enter image description here

diverges?

Using symbolab gave me diverges, but it does not show how, and it used the series root test, which I will not cover in my course.

Thank you.

user151408
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    The sequence $ 2/ (3n + 1)$ doesn't diverge. Do you mean $$\sum_{n = 1}^{\infty} \frac{2}{3n + 1} ?$$ If so, do you know that the harmonic series diverges? –  May 31 '14 at 02:49
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    Compare with $\sum \frac{2}{4n}$, or use Integral Test. – André Nicolas May 31 '14 at 02:50

2 Answers2

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(Limit) Comparison test with $\sum_{n=1}^\infty \frac{1}{n}$, which diverges.

Paul Hurst
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$\dfrac{2}{3n+1} > \dfrac{2}{4n} = \dfrac{1}{2}\cdot \dfrac{1}{n}$. You can compare your series with the one on the right: $\displaystyle \sum_{n=1}^\infty \dfrac{1}{n}$. Thus your actual question is how to show this harmonic series diverges. Let $a_n = \displaystyle \sum_{k=1}^n \dfrac{1}{k}$. You can show $a_n \to +\infty$. Compare this sequence with $b_n = \text{lnn}$. You can prove by integration or induction that $a_n > b_n$, and $b_n \to +\infty$, so $a_n \to +\infty$, and you are done.

Alternately: you can prove that: $a_{2^n} \geq 1 + \frac{n}{2} \to +\infty$. So:

$a_{2^n} \to +\infty$, this implies: $a_n \to +\infty$.

DeepSea
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