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This is basically a question based on Problem 6(b) of Chapter 23 in Spivak's Calculus. I have to prove that if the following are met:

  • $f$ is continuous on an interval around $0$
  • $a_n=f\left(\frac{1}{n}\right)$ (for n large enough)

  • $\sum\limits_{n=1}^\infty a_n$ converges

  • $f'(0)$ exists

    then $f'(0)=0$

I have already proven in part (a) that $f(0)=0$ and for part (b) I tried to work like this: $f'(0)= \lim\limits_{x\to 0}\frac{f(x)-f(0)}{x-0}=\lim\limits_{n\to \infty }nf(1/n)=\lim\limits_{n\to \infty }na_n $.

I tried to calculate $\lim\limits_{n\to \infty }na_n $ by claiming that $a_n$ is bounded(which I think is correct) but I seem to be missing something.

Any help?

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HINT: You're told that $f'(0)$ exists. Suppose it is $L\ne 0$. Using what you've said, what does that tell you about $a_n$ for large $n$?

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