# If the series $\sum_{n\ge 1}f_n \left(x \right)$ converges but not uniformly does the sequence $\left(f_n\left(x\right)\right)$ converge?

From Cauchy Criterion for Uniform Convergence we can conclude that if the series $\sum_{n\ge 1}f_n \left(x \right)$ converges uniformly than the sequence $\left(f_n\left(x\right)\right)$ converges uniformly to $f\left(x\right)=0$.

But if the series $\sum_{n\ge 1}f_n \left(x \right)$ converges but not uniformly does it mean that the sequence $\left(f_n\left(x\right)\right)$ must also converge ?

I failed to prove this statement and failed to found counter-example of convergent but not uniformly convergent series with divergent sequence.

Could you please give me some hint ?

Thanks.

• If a sum converges, it terms go to zero. – Pedro Tamaroff Jan 9 '14 at 7:44
• Of course ! Nice and simple. – user97484 Jan 9 '14 at 8:03

If the series $\sum_n f_n(x)$ converges pointwise, then the sequence $f_n(x)$ converges pointwise to zero. To see this, fix the variable $x$. Then the question simply becomes one about sequences and series of numbers rather than sequences and series of functions.