<s>[INST] $\lim_{N\to \infty} \sum_{n=N}^{2N} c_n = 0$ $\Rightarrow \sum_{n=1}^{\infty} c_n$ converges? If $\lim_{N\to \infty} \sum_{n=N}^{2N} c_n = 0$ do we have that $\sum_{n=1}^{\infty} c_n$ converges? At first this did not seem true($\sum_{n=N}^{2N} (-1)^n$ is $0$ when N is odd), but I've failed to find a proper counter-example. I've tried thinking in terms of partial sums to prove that is it true but have had no luck. Can someone help?  
 [/INST] Let 
\begin{align*}
c_1 = c_2 &= \frac{1}{2}\\
& \\
c_3 = c_4 = c_5 = c_6 &= \frac{1}{4}\times\frac{1}{2} = \frac{1}{8}\\
& \\
c_7 = c_8 = c_9 = c_{10} = c_{11} = c_{12} = c_{13} = c_{14} &= \frac{1}{8}\times\frac{1}{3} = \frac{1}{24}
\end{align*}
and so on. More precisely, let $c_n = \dfrac{1}{2^kk}$ where $k = \lfloor\log_2(n+1)\rfloor$. 
Then if $S_m = \displaystyle\sum_{n=1}^ma_n$, we have 
$$S_{2^{p+1}-2} = \sum_{n=1}^{2^{p+1}-2} c_n = \sum_{k=1}^{p}\sum_{n=2^k-1}^{2^{k+1}-2}c_n = \sum_{k=1}^{p}\sum_{n=2^k-1}^{2^{k+1}-2}\frac{1}{2^kk} = \sum_{k=1}^{p}2^k\frac{1}{2^kk} = \sum_{k=1}^p\frac{1}{k}.$$
As the harmonic series diverges, $S_{2^{p+1}-2}$ diverges. Therefore $\displaystyle\sum_{n=1}^{\infty}c_n$ diverges.
</s>