Dear Uncle Colin,

Is there an easy way to write series in sum notation? I have $1 + \frac{1}{2\sqrt{2}}+\frac{1}{3\sqrt{3}} + …$ but no real clue about how to put it into a compressed form.

- Summing Is Giving Me Aneurysms

Hi, SIGMA, and thanks for your message!

I like to make a little table for these, especially if it’s not obvious what’s going on. Then I can ask myself what’s changing and what’s staying the same.

Here, I’d start:

Term Value
1 $1$
2 $\frac{1}{2\sqrt{2}}$
3 $\frac{1}{3\sqrt{3}}$
4 $\frac{1}{4\sqrt{4}}$
… …

If you write the first term as $\frac{1}{1\sqrt{1}}$, you can see exactly what changes each time: the two numbers on the bottom of the fraction are each the same as the number of the term, suggesting the $n$th term is $\frac{1}{n\sqrt{n}}$. I’d probably write that as $n^{-\frac{3}{2}}$, giving the final answer of $\Sigma_1^\infty n^{-\frac{3}{2}}$.

Hope that helps!

-- Uncle Colin

* Updated 2017-01-11 to make a $\sigma$ a $\Sigma$. Thanks, @robjlow!