For what values of $c$ will $\sum_{n=1}^{\infty}\left(\frac{c}{n} - \frac{1}{n+1}\right)$ converge? (verify solution)
Solution 1:
You might pave the way differently. We know that if $p>1$ and $$\displaystyle\lim_{n\to\infty}n^pu_n<\infty$$ then $\sum u_n$ converges. This, for the question, is equivalent to have $\displaystyle\lim_{n\to\infty}n^{p-1}(c-1)<\infty$, so it should be $c=1$ if we are looking for the convergence!