# Convergence of a functional series (analysis)

1. Dec 16, 2012

### mred11

1. The problem statement, all variables and given/known data

Determine whether the following functional series is pointwise and/or uniformly convergent:

$\sum_{n=1}^\infty \frac{x}{n} (x\in\mathbb{R})$

2. Relevant equations

3. The attempt at a solution

My answer to this seems very straightforward and I would be very grateful if someone could let me know if the method is correct. We can re-write the series as:

$x \sum_{n=1}^\infty \frac{1}{n}$

We know the harmonic series (of numbers) $\sum_{n=1}^\infty \frac{1}{n}$ is divergent, so therefore the series cannot be pointwise or uniformly convergent.

Is this correct? Thanks!

2. Dec 16, 2012

### sharks

You could also use the p-series test to arrive at the same conclusion.

3. Dec 16, 2012

### mred11

Thanks. I just noticed something: if x=0, then the series is just zero, so it converges. Given that the series isn't convergent for *all* values of x in the domain, is it still correct to say the functional series is divergent? I suppose it's a question of terminology. Thanks for the help.

4. Dec 16, 2012

### pasmith

It's pointwise convergent at points where it converges, and pointwise divergent where it doesn't converge.

5. Dec 16, 2012

### sharks

Based on your question, i would say that the correct answer is: point-wise convergent at x = 0. (which would imply that at any other values of x, the series is divergent)

6. Dec 17, 2012

Ok, thanks!