# Series

The sum of a sequence of terms is called a series.

Let $$(a_n)_{n\geq k}$$ be a sequence starting with index $$k$$. Now sum over the first $$n-k-1$$ terms:

$\begin{array}{rl}s_n =& \sum\limits_{i=k}^n a_i\\=& \underbrace{\underbrace{\underbrace{\underbrace{a_k}_{s_k} + a_{k+1}}_{s_{k+1}}+a_{k+2}}_{s_{k+2}}+...+a_n}_{s_n}\end{array}$

The sequence $$(s_n) = (s_k, s_{k+1}, s_{k+2}, ...)$$ is called an infinite series $$\sum\limits_{i=1}^\infty a_i$$. If $$(s_n)$$ converges, its limit is called $$\sum\limits_{i=k}^\infty a_i$$.

## Examples

• $$\sum\limits_{i=1}^\infty i = 1+2+3+...$$ diverges
• $$\sum\limits_{i=1}^\infty (-1)^i$$ diverges, $$s_n = \sum\limits_{i=1}^n(-1)^i = \begin{cases} 0, & \text{if } n \text{ even} \\ -1, & \text{if } n \text{ odd} \end{cases}$$
• Harmonic series $$\sum\limits_{i=1}^\infty\frac{1}{i}$$ diverge, because new packages that are $$\geq\frac{1}{2}$$ can always be put together:
$$\begin{array}{rl} s_n &= 1+\underbrace{\frac{1}{2}+\frac{1}{3}}_{>\frac{1}{2}}+\underbrace{\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}}_{>\frac{1}{2}}+...+\frac{1}{n}\\&<1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+...\end{array}$$
• $$\sum\limits_{i=1}^\infty\frac{1}{i^2}$$ converges to $$\frac{\pi^2}{6}$$
• Geometric Series: $$\sum\limits_{i=0}^\infty q^i \begin{cases} |q|<1: \text{ converges to }\frac{1}{1-q} \\ |q|\geq 1: \text{ diverges} \end{cases}$$
• General Harmonic series: $$\sum\limits_{i=1}^\infty\frac{1}{i^s}$$ converges $$\forall s>1$$
• Leibnitz-Series: $$\sum\limits_{i=0}^\infty(-1)^i\frac{1}{2i+1}$$ converges to $$\frac{\pi}{4}$$
• Alternating Harmonic series: $$\sum\limits_{i=0}^\infty(-1)^i\frac{1}{i+1}$$ converges to $$\ln 2$$
• $$\sum\limits_{i=1}^\infty\frac{1}{2^i}$$ converges to $$1$$ since

We can state that if $$\sum\limits_{i=k}^\infty a_i$$ converges $$\Rightarrow (a_i)_{n\geq k}$$ is a null sequence. $$\Leftarrow$$ does not hold, see Harmonic series.

## Convergence / Divergence criteria

### Divergence criterion

If $$(a_n)_{n\geq k}$$ is no null sequence, the series $$\sum\limits_{i=k}^\infty a_i$$ diverges.

Example: $$\sum\limits_{i=1}^\infty\left(1+\frac{1}{i}\right)$$ is divergent, since $$\lim\limits_{i\to\infty}\left(1+\frac{1}{i}\right)=1$$

### Weierstrass comparison test (majorant criterion)

Let $$(a_n)_{n\geq k}$$, $$(b_n)_{n\geq k}$$ be sequnces with $$|a_n|\leq b_n$$, then: If $$\sum\limits_{i=k}^\infty b_i$$ is convergent then $$\sum\limits_{i=k}^\infty |a_i|$$ is also convergent as well as $$\sum\limits_{i=k}^\infty a_i$$.

### Leibniz test for alternating series

Let $$(a_n)_{n\geq k}$$ be a monotonically decreasing null sequence with $$a_i\geq 0\forall i$$, then the alternating series $$\sum\limits_{i=k}^\infty (-1)^ia_i$$ converges.

### Absolute Convergence

$$\sum\limits_{i=k}^\infty a_i$$ is called absolute convergent, if $$\sum\limits_{i=k}^\infty|a_i|$$ converges. It is true that if a series converges absolute $$\Rightarrow$$ the series converges. $$\Leftarrow$$ does not hold, see alternating Harmonic series.

### Root test

If there exists a $$q<1$$ and an index $$i_0$$ for which $$\sqrt[i]{|a_i|}\leq q\forall i\geq i_0$$ holds, the series $$\sum\limits_{i=k}^\infty a_i$$ converges absolute.

Please note: $$\sqrt[i]{|a_i|}< 1$$ is not enough! For example the Harmonic series $$\sqrt[i]{\frac{1}{i}}\to 1$$, but we can’t find a $$q<1$$.

If $$\sqrt[i]{|a_i|}\geq 1$$ holds for endless $$i$$, the series diverges.

### d’Alembert’s ratio test (quotient criterion)

If there exists a $$q<1$$ and an index $$i_0$$ for which $$\left|\frac{a_{i+1}}{a_i}\right|\leq q\forall i\geq i_0$$ holds, the series $$\sum\limits_{i=k}^\infty a_i$$ converges absolute.

Please note: $$\left|\frac{a_{i+1}}{a_i}\right|< 1$$ is not enough!

For $$\left|\frac{a_{i+1}}{a_i}\right|\geq 1$$ no general statement is possible.

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