Question

a) Suppose we know that the series $\sum_{n=1}^{\infty} a_{n}$ is convergent, where the sequence a_n is nonzero. Show that the series $\sum_{n=1}^{\infty} \frac{1}{a_{n}}$ is divergent by applying the appropriate test.

b) Suppose we know that the series $\sum_{n=1}^{\infty} c_{n}$ is convergent, where the sequence c_n consists of exclusively positive terms. Show that the series $\sum_{n=1}^{\infty} \frac{c_{n}}{n}$ is convergent by applying the appropriate test.  

Answer 1

Sold) Lim an=0 O Since Į an is n=1 is convergent no Lem I ol- - 2 to - π an Liman no n - a So by the divergent test Ean is divergent. n=1 le Since E cn es con convergent and cn >0, then n=1 So, by the companson test- Como cal Since → Co is convergent. Een es convergent n=1 n!