# A series of functions that converges pointwise

R.P.F.

## Homework Statement

Hi,

Can someone give me an example where f_n is continuous on [0,1] for each n.
$$f = \sum_{n=0}^\infty f_n$$ converges pointwise(not uniformly) on [0,1] and f is not continuous on [0,1]?

Thanks!

## The Attempt at a Solution

Technically we need uniform convergence, but I am having trouble coming up with such an example.

## Answers and Replies

Homework Helper
A typical example is the sequence of functions fn(x) = x^n on [0, 1]. It converges pointwise, but not uniformly. And its limit f is not continuous.

R.P.F.
A typical example is the sequence of functions fn(x) = x^n on [0, 1]. It converges pointwise, but not uniformly. And its limit f is not continuous.

Hey sorry not to be clear. I meant that the infinite sum converges. So I'm talking about a series of functions, not a sequence of functions.

Homework Helper
Ah yes, sorry, I was being a bit harsh.

An example which should serve is the series $$f_{n}(x)=\frac{x^2}{(1+x^2)^n}$$, on an interval containing 0.

R.P.F.
Ah yes, sorry, I was being a bit harsh.

An example which should serve is the series $$f_{n}(x)=\frac{x^2}{(1+x^2)^n}$$, on an interval containing 0.

I see. Thanks!