How can we prove that f'(0) = 0 if the series converges

[ptolema]

Homework Statement

let f be a continuous function on an interval around 0, and let a_n=f(1/n) (for large enough n)
prove that if f'(0) exists and

converges, then f'(0)=0

Homework Equations

proved earlier in the problem that if the series converges, then f(0)=0.

The Attempt at a Solution

for this part of the problem, i found that

by l'Hopital, so f' is continuous at 0. how can i then draw the conclusion that f'(0)=0? i know that the series should tie in somehow, but I'm getting stuck on making the connection.

 

[Dick]

Call c=f'(0). Then you know c=lim n->inf f(1/n)/(1/n). So for large n, f(1/n) is 'like' c/n. Does the series c/n converge?

 

[ptolema]

ok, so i just did the work to get to that part. so for large n, f'(0)/n is approx f(1/n), and f(1/n) is approx 0. since f'(0)/n is "like" 0, f'(0) approaches 0 for large n. is that something like what you were saying?

 

[Dick]

ok, so i just did the work to get to that part. so for large n, f'(0)/n is approx f(1/n), and f(1/n) is approx 0. since f'(0)/n is "like" 0, f'(0) approaches 0 for large n. is that something like what you were saying?

Not at all! f'(0)/n may approach zero for large n, but that's because n->infinity! It doesn't say anything at all about f'(0). The hint I'm trying to give you is that the series c/n does not converge if c is not zero.