Exercise 8.3.7

We showed in Exercise 2.7.10 (b) that $\lim <!--\; FUNCTION\; APPLICATION\; -->c_n=0$. We can show that ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{n=0}^{\infty }{c}_n$ diverges by comparison against the harmonic series $(h_n)$, which we modify for convenience to define $h_n=1∕(n+1)$. We have $c_0\ge h_0$, and note that

for $n\ge 1$, so by induction $c_n\ge h_n$.