Convergence Proof for Continuous Functions with Second Derivative at Zero

[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 f'(0)=f(0)=0, then

converges

Homework Equations

i proved earlier in the problem that if the series converges, f(0)=0, and if f'(0) exists and the series converges, then f'(0)=0

The Attempt at a Solution

not really sure how to approach this. don't know if a convergence test should be used, but i know that lim n->infinity a_n=0 (zero test is inconclusive). what should i do?

 

[lanedance]

so you know
f(0) = 0

f'(0) = lim x->0, f(x)/x = 0

f''(0) = lim x->0, f'(x)/x = a

the 2nd line shows, f(x) tends to zero faster than x...

 

[ptolema]

i'm not really sure how that helps. that takes me back to the result of the zero test, and the fact that f'(0)=0, which i already proved. how does that pertain to the series.

 

[lanedance]

how about working towards a comparison test...? as f(x)/x tends to zero, near 0, maybe you can do something with f(1/n)/(1/n)