Estimate the sum of alternating harmonic series between $7/12$ and $47/60$

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 $47/60$.

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.