Series

A sequence is an ordered collection of numbers. We can, for instance, define a sequence of square numbers $(s_i{)}_i$ as $s_i:=i^2$.

Using this definition, we can define the next interesting concept:

Definition: Series

A series is the sequence of partial sums of a sequence. The series $(t_n{)}_n$ of the sequence $(s_i{)}_i$ is defined as

$$t_n:=\sum _{i=1}^n{s}_i.$$

Natural numbers

A trivial sequence is formed by the natural numbers themselves, $s_i=i$. The series is interesting, because each term is the sum of $n$ natural numbers:

$$S_n=\sum _{i=1}^{n}i.$$

There is a clever way to compute this for arbitrary $n$ in closed form:

Theorem: Gauss's Formula

The value of $S_n$ is given by

$$S_n=\frac{n(n+1)}{2}.$$

Proof

Writing out the sum, we have

$$S_n=1+2+\cdots +(n-1)+n.$$

As summation is commutative, we can also write this series out in reverse:

$$S_n=n+(n-1)+\cdots +2+1.$$

Both sums have the same number of terms and the same value. We can add both lines and match up the element:

$$2S_n=(n+1)+(n+1)+\cdots +(n+1)+(n+1).$$

We have $n$ elements which are $(n+1)$. Hence we can simplify to

$$2S_n=n(n+1).$$

We divide by $2$ to isolate $S_n$ to get the claimed result:

$$S_n=\frac{n(n+1)}{2}.$$