Example of a series!

Let:

• $a_{3n} = \frac{1}{n+1}$
• $a_{3n+1} = 0$
• $a_{3n+2} = \frac{-1}{n+1}$

Then $\sum a_{3n}$ is the harmonic series and diverges.

Let $S_n=\sum_{k=0}^n a_k$, then, $\forall n\in \mathbb{N}, S_{3n} = \frac{1}{n+1}; S_{3n+1}=S_{3n+2}=0$ (try and prove it by induction if needed).

So, $\forall n \in \mathbb{N},0\leq S_n \leq \frac{1}{n+1}$ and by the squeeze theorem, since $\lim_{n\to \infty} \frac{1}{n+1}=0$, the series converges to $0$.

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On the other hand, let:

• $a_{3n} = \frac{1}{n^2}$
• $a_{3n+1} = 1$
• $a_{3n+2} = 1$

Then $\sum_{k=1}^{3n} a_k \geq 2n$ diverges but $\sum a_{3n}$ converges as a Riemann series.