# Help with proving that this series diverges

I have this complex series $\sum\limits_{n=1}^{\infty} \frac{i^n}{\sqrt{n}}$, which I'm trying to prove converges.

Now, I know that a complex sequence converges iff both its real and its imaginary partial sums converge.

Obviously, $\sum\limits_{n=1}^{\infty} \Im(a_n)=\sum\limits_{n=1}^{\infty}0 \to 0$.

But, when I try to compute $\sum\limits_{n=1}^{\infty} \Re(a_n),$ I end up with $\sum\limits_{n=1}^{\infty}\frac{1}{\sqrt{n}},$ which, whilst, intuitively, I know diverges, I can't seem to prove. I know I have to use the Comparison Test, but to what should I compare it?

By the way, 'my' version of the Comparison Test is as follows (for divergence): If $\frac{b_n}{a_n}$ is bounded above and if $\sum\limits_{n=1}^{\infty} b_n$ diverges, then $\sum\limits_{n=1}^{\infty}a_n$ diverges.

Could someone give me a hint?

Thanks

• Use the integral test to show that your series in question diverges. That's fairly easy to integrate – imranfat May 9 '14 at 14:58
• You have a sign error. The signs should alternate. – André Nicolas May 9 '14 at 14:58
• It will be easier if you use Dirichlet's test instead. – achille hui May 9 '14 at 14:58
• Sorry, I should mention that I have to use the Comparison Test; we haven't done either of these suggested tests. – beep-boop May 9 '14 at 14:59
• You have done it, it was probably called the Alternating Series test. – André Nicolas May 9 '14 at 15:00

Obviously, $\sum\limits_{n=1}^{\infty} \Im(a_n)=\sum\limits_{n=1}^{\infty}0 \to 0$.

This is not the case.

As André Nicolas suggests in the comments, it is best to write out the first few terms to get a feel for the series: $$\frac{i}{\sqrt1} + \frac{-1}{\sqrt{2}} + \frac{-i}{\sqrt3} + \frac{1}{\sqrt4} + \frac{i}{\sqrt5} + \frac{-1}{\sqrt6} + \frac{-i}{\sqrt7} + \frac{1}{\sqrt9} + \frac{i}{\sqrt{10}} + \cdots$$ You correctly observe that you should look at the real and imaginary parts. \begin{align*} \sum \Im (a_n) &= \frac{1}{\sqrt{1}} - \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{5}} - \frac{1}{\sqrt{7}} + \cdots \\ \sum \Re (a_n) &= -\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{4}} - \frac{1}{\sqrt{6}} + \frac{1}{\sqrt{8}} - \cdots \\ \end{align*}

The proper way to show these converge is with the alternating series test.

If you were trying to prove that the real and/or imaginary partial sums diverge (by the Alternating Series Test), could you just take the odd subsequence and show that that diverges?

The alternating series test requires these two conditions:

1. The terms alternate in sign.

2. The terms decrease in absolute value.