Theory

Theory 1

A series is an infinite sum that is created by successive additions without end. The terms are not added up “all at once” but rather they are added up “as $n$ increases” or “as $n\to \infty$.”

$$a_0+a_1+a_2+a_3+\dots =\sum _{n=0}^{\infty }{a}_n$$

Three of the most famous series are the Leibniz series and the geometric series:

$$\begin{array}{cccc}\text{Leibniz series:}& 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots +\frac{{(-1)}^n}{2n+1}+\cdots & =& \frac{\pi }{4}\\ & & & \\ \text{Geometric series:}& 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots +{\left(\frac{1}{2}\right)}^n+\cdots & =& 2\end{array}$$

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Partial sum sequence of a series

The partial sum sequence of a series is the sequence whose terms are the sums up to the given index:

$$S_N=a_0+a_1+\cdots +a_N=\sum _{n=0}^N{a}_n$$

These $S_N$ terms themselves form a sequence:

$$S_0,S_1,S_2,S_3,\dots$$