Estimate the sum of alternating harmonic series between $7/12$ and $47/60$ How can I prove that: 
$$\frac{7}{12} < \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} < \frac{47}{60}$$ 
? I don't even know how to start solving this... 

If $s_{n}:=\sum_{k=1}^{n}\frac{\left(-1\right)^{k+1}}{k}$ then $\left(s_{2n-1}\right)_{n=1,2,\dots}$
is monotonically decreasing and $\left(s_{2n}\right)_{n=1,2,\dots}$
is monotonically increasing. 
This with $s_{2n}<s_{2n-1}$. 
So if you can find some $n$ with $\frac{7}{12}<s_{2n}<s_{2n-1}<\frac{47}{60}$
then you are ready.
