Ask Uncle Colin: How do I add things up?

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to and Uncle Colin will do what he can.

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!


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


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