<s>[INST] showing $ 1-\frac{3}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{3}{6}+\frac{1}{7}+\frac{1}{8}+\cdots=0 $ How to show that the following infinite series
$$
1-\frac{3}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{3}{6}+\frac{1}{7}+\frac{1}{8}+\cdots=0?
$$
The above series is of the form $\sum_{n \ge 1} \frac{f(n)}{n}$, where $f$ is a periodic arithmetical function of period $4$, with the values $f(1)=f(3)=f(4)=1$ and $f(2)=-3$. Since $\sum_{1 \le i \le 4} f(i)=0$, it is assured that this series is convergent.
 [/INST] sum of all odd terms: $X = (1 + 1/3 + 1/5 + ... )$
sum of all terms (4n+2): $Y = -3/2(1 + 1/3 + 1/5 + ...) = -3X/2$
Sum of remaining terms = $Z = 1/4 + 1/8 + 1/12 + ... = 1/4(1 + 1/2 + 1/3 + 1/4 + 1/5 + ...)$
splitting into odd and even terms: $Z = 1/4(1+ 1/3 + 1/5 +...) + 1/8(1 + 1/2 + 1/3 + ...)$
So $Z = X/4 + Z/2 \implies Z = X/2$
So $X + Y + Z = X - 3X/2 + X/2 = 0$
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