Can the following limits be zero?

Question

$\underset{k\to \infty }{\mathop{\lim }}\,\sum\limits_{i=1}^{k}{{{a}^{k-i}}{{e}_{p}}(i+1)} = 0 ?$

where |a| <1, and $\underset{k\to \infty }{\mathop{\lim }}\,{{e}_{p}}(k)=0 $.

Answer 1

Hint

• $\lim_{k\to \infty} e_p(k)=0\implies |e_p(k)|$ is bounded.
• $\displaystyle \sum_{i=1}^k a^{k-i}=\sum_{j=0}^{k-1} a^j.$