Convergence of a functional series (analysis)

[mred11]

Homework Statement

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

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

Homework Equations

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!

 

[DryRun]

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

 

[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.

 

[pasmith]

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?

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

 

[DryRun]

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

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)

 

[mred11]

Ok, thanks!