Infinite Series

analysis

Definition

Infinite Series

An infinite series is the sum of an infinite sequence of terms.

Let $(a_n{)}_{n\ge 0}$ be a sequence.

$$\sum _{n=0}^{\infty }{a}_n=\underset{n\to \infty }{\lim }\underset{s_k}{\underbrace{\sum _{k=0}^n{a}_k}}$$

where $s_k$ is called the partial sum sequence of that infinite series.

Commutativity and Associativity

For infinite series, neither commutativity nor associativity can be applied.

Example:

$$1-1+1-1+\dots \ne \underset{=0}{\underbrace{(1-1)+(1-1)+\dots }}\ne \underset{=1}{\underbrace{1-(1-1)-(1-1)-\dots }}$$

Partial Sum Sequence

Partialsummenfolge

Since we can’t apply commutativity and associativity, dealing with infinite series becomes quite hard. Thus, we use a partial sum sequence, which is nothing more than the infinite sum but limited to the point $n$.

The partial sum of sequence sequence $(a_n{)}_{n{\ge }_0}$ is given by:

$$s_n=\sum _{k=0}^n{a}_k$$

Since a partial sum is finite, we can apply commutativity and associativity, i.e. telescope sum.

Convergence

A infinite series is called convergent iff its partial sum sequence is convergent:

$$\sum _{n\ge 0}{a}_n\text{ convergent}⟺(s_n{)}_{n{\ge }_0}\text{ convergent}$$

Convergence and Zero Sequence

Given a sequence $(a_n{)}_{n\in \mathbb{N}}$ and an infinite series ${\sum }_{n\in \mathbb{N}}{a}_n$. The infinite series is convergent implies that $(a_n{)}_{n\in \mathbb{N}}$ is a zero sequence.

$$\begin{array}{rl}{a}_n& =s_n-s_{n-1}=\sum _{k=0}^n{a}_n-\sum _{k=0}^{n-1}{a}_n\\ \underset{n\to \infty }{\lim }{a}_n& =\underset{n\to \infty }{\lim }{s}_n-\underset{n\to \infty }{\lim }{s}_{n-1}\\ & =0\end{array}$$

Equivalently, an infinite series is called divergent iff its partial sum sequence is divergent:

$$\sum _{n\ge 0}{a}_n\text{ divergent}⟺(s_n{)}_{n{\ge }_0}\text{ divergent}$$

Zero Sequence Convergence

Value

If an infinite series is convergent, then the value of the infinite series is given by:

$$\sum _{n\ge 0}{a}_n:=\underset{n\to \infty }{\lim }{s}_n$$

Lecture

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