It seems like we need a better way of writing infinite sums that doesn't depend on guessing patterns. Luckily, there is one. It's easiest understood using an example: $$\sum_{i=1}^\infty \frac{1}{i}.$$ The symbol $\sum$ stands for "sum". The symbol $i$ is a dummy variable. The formula tells us to form a sum whose terms are the expression that comes after the $\sum$ with the symbol $i$ replaced by $1$, $2$, $3$, $4$, and so on, all the way to infinity. Thus, $$\sum_{i=1}^\infty \frac{1}{i}$$ stands for $$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + ... .$$
The expression $$\sum_{i=1}^\infty i$$ stands for $$1+2+3+4+5...$$ and $$\sum_{i=1}^\infty \frac{1}{i^2}$$ stands for $$1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + ... .$$ You can even express sums whose terms are alternatingly negative and positive, for example $$\sum_{i=1}^\infty (-1)^n \frac{1}{i}$$ stands for $$-1 + \frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + ... .$$ What about our sum from above, $$1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{6} - \frac{1}{8} + \frac{1}{5} - ...?$$ Can you express it in terms of the $\sum$ notation? It's a little tricky, so if you can't find the answer, see here.
But why would we want to write down infinite sums in the first place? The reason is that even though they are infinitely long, they can still converge to a finite value. In fact, writing certain quantities as infinite sums (more properly called infinite series) is a powerful tool in mathematics. See here to find out more about infinite series.
Comments
Confused!
I didn't understand the answer for the sum you mentioned, the tricky one... can you elaborate?
sum in the big sigma
The presentation is a bit weak. The intention is that all three fractional values are included within the scope of the big sigma. There ideally would be a big bracket around all the terms to show that they are inside the big sigma summation, but that level of detail is often sadly omitted.
For each step of the index dummy variable you get three fractional values. I hope this helps.
Typo in the last Big Sigma in the article
The index variable is i but inside the sigma we have (-1)^n. It might be difficult to read with (-1)^ i so it might be better to change the index variable to r or n.