Bois-Reymond criterion for series

[mahler1]

Homework Statement .

Prove that if $\sum_{n=1}^{\infty} (a_n-a_{n-1})$ converges absolutely and $\sum_{n=1}^{\infty} z_n$ converges, then $\sum_{n=1}^{\infty} a_nz_n$ converges.

The attempt at a solution.

I know that if $Z_N=z_0+z_1+...+z_N$, then $\sum_{n=0}^N a_nz_n= a_NZ_N-\sum_{n=0}^{N-1} Z_n(a_{n+1}-a_n)$

I am not so sure how can I use the hypothesis given to this new expression or if it would be more convenient to express the original series in another way.

 

[xiavatar]

Your formula $\sum_{n=0}^N a_nz_n= a_NZ_N-\sum_{n=0}^{N-1} Z_N(a_{n+1}-a_n)$ is incorrect.

$a_NZ_N-\sum_{n=0}^{N-1} Z_N(a_{n+1}-a_n)=0$.

 

[mahler1]

Your formula $\sum_{n=0}^N a_nz_n= a_NZ_N-\sum_{n=0}^{N-1} Z_N(a_{n+1}-a_n)$ is incorrect.

$a_NZ_N-\sum_{n=0}^{N-1} Z_N(a_{n+1}-a_n)=0$.

I've corrected it