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... 

First note that the series converges using Leibniz Test.
Next, denote by $S_N$ the partial sum $\sum_{n=1}^N\frac{(-1)^{n-1}}{n}$.  Then, we must have $$S_{2N}<\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}<S_{2N+1}$$
Finally, we see that $\sum_{n=1}^4 \frac{(-1)^{n+1}}{n}=\frac{7}{12}$ and inasmuch as the next term is positive, the value of the series must exceed $7/12$.  Similarly, we see that $\sum_{n=1}^5 \frac{(-1)^{n+1}}{n}=\frac{47}{60}$ and inasmuch as the next term is negative, the value of the series must be less than $47/60$.
