Maths in a minute: Writing infinite sums

If I tell you to consider the infinite sum $$1+2+3+4+5...$$ you will know what I mean. It's easy to continue the pattern, it's 6 next, then 7, and so on. The same goes for the infinite sum $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...$$ But what if the pattern is less obvious? For example, it's a little harder to see how to continue

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: $$\sum _{i=1}^{\mathrm{\infty }}\frac{1}{i}.$$ 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, $$\sum _{i=1}^{\mathrm{\infty }}\frac{1}{i}$$ stands for $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+....$$

The expression $$\sum _{i=1}^{\mathrm{\infty }}i$$ stands for $$1+2+3+4+5...$$ and $$\sum _{i=1}^{\mathrm{\infty }}\frac{1}{i^2}$$ 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 $$\sum _{i=1}^{\mathrm{\infty }}(-1{)}^n\frac{1}{i}$$ 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.