# Pointwise or uniform convergence of $\sum_{n=1}^{\infty}e^{i\sqrt{n}x}/n^2$

We need to tell about this series $$\sum_{n=1}^{\infty}\frac{e^{i\sqrt{n}x}}{n^2}, x\in\mathbb{R}$$

that if it converges pointwise or uniformly on $\mathbb{R}$?

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Did I get the title right in my edit? – Argon Jul 28 '12 at 22:16
Weierstrass M-test ... – Matt Jul 28 '12 at 22:23
This looks like homework; please read meta.math.stackexchange.com/questions/1803/…;. – Nate Eldredge Jul 28 '12 at 22:27
What did you try? – Did Jul 28 '12 at 22:29

## 1 Answer

There's something interesting happening here, which you may not have seen before.

What is the difference between exp(x) and exp(ix), where x is real? Do both of them blow up as x grows towards infinity? If you understand the difference between the exponential function with real exponent and imaginary exponent, this question will be simple.

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Hint to go along with this answer and my comment about the M-test. Write out the formula for $e^{a+ib}=e^ae^{ib}$ using Euler's formula. From this you can derive a general formula for $|e^{a+ib}|$ which will answer the question in this answer...or maybe the poster is never coming back. – Matt Jul 29 '12 at 2:59