[Math] Find all values of $c$ for which $\sum_{n=1}^{\infty} \left(\frac{c}{n}-\frac{1}{n+1}\right)$ converges

Question

Find all values of $c$ for which $\sum\limits_{n=1}^{\infty} \left(\frac{c}{n}-\frac{1}{n+1}\right)$ converges.

I know that the series in question converges when $c=1$, but I have no concrete way to find all such values of $c$ for which this is true.

Answer 1

Write the summands as

$$\left[\frac{1}{n}-\frac{1}{n+1}\right]+\frac{c-1}{n}.$$

With this observation, it's easy to see that

$$\sum_{n=1}^N\left(\frac{c}{n}-\frac{1}{n+1}\right)=1-\frac{1}{N+1}+(c-1)\sum_{n=1}^N\frac{1}{n}.$$

We know the harmonic series diverges, so this converges iff $c=1$.