Suppose that an ≥0 and that ∑n=1^∞ an

Question

Suppose that an ≥0 and that ∑n=1^∞ an<∞. (i) Show that lim infn →∞ n an=0. (ii) Give an example

Accepted Answer

Step 1: We start with the given condition that u005C( u005Csum_{nu003D1}^{u005Cinfty} a_n u003C u005Cinfty u005C). This mean

Suppose that $a_n \geq 0$ and that $\sum_{n=1}^{\infty} a_n<\infty$. (i) Show that $\liminf _{n \rightarrow \infty} n a_n=0$. (ii) Give an example showing that $\lim \sup _{n \rightarrow \infty} n a_n>0$ is possible.