# Series

## Definitions #

Infinite sequence: function of which the domain is the set of positive integers and the range is a set of real numbers

$$\{a_n\}_{n=1}^\infty \text{ or } \{a_n\}$$

Infinite series: the sum of an infinite sequence $$\sum_{k=1}^\infty a_k \text{ or } \sum a_k$$

Partial sum: $$S_n = a_1 + a_2 + a_3 + ... + a_n = \sum_{k=1}^n a_k$$

## Convergent/divergent series #

• Convergent: If $\lim_{n \to \infty} a_n = L$ does exsist

• Divergent: If $\lim_{n \to \infty} a_n = L$ does not exsist

If $S_n$ is convergent then series is convergent and has sum: $$\lim_{n \to \infty} S_n = \sum_{k=1}^\infty a_k = S$$ Otherwise, if the partial sum is divergent then the serries is divergent.

## Special series #

### Geometric series #

$$\sum_{n=1}^\infty a r^{n-1} = a + ar + ar^2 + ...$$

• Convergent: $|r| < 1$ with sum $S = \frac{a}{1-r}$
• Divergent: $|r| \geq 1$

### Harmonic series #

$$\sum_{n=1}^\infty \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n} + ...$$

$\lim_{n \to \infty} \frac{1}{n} = 0$

## Tests #

### nth term test #

If $\sum_{n=1}^\infty a_n$ is convergent, then $\lim_{n \to \infty} a_n = 0$. (only in this order)

If $\lim_{n \to \infty} a_n \neq 0$ or does not exsist, the series is divergent.

Note that if $\lim_{n \to \infty} a_n = 0$, you can't conclude that the series converges!

### Integral test #

If $a_k = f(k)$ for all positive integers $k$ and $n$ is any positive integer (often 1) then $\sum_{k=n}^\infty a_k$ converges if and only if $\int_n^\infty f(x) dx$ exsists.

### Ordinary/direct comparison test #

Suppose $0 \leq a_n \leq b_n$:

• If $\sum b_n$ converges, then $\sum a_n$ converges
• If $\sum a_n$ diverges, then $\sum b_n$ diverges

### Limit comparison test #

$$\lim_{n \to \infty}\frac{a_n}{b_n}=c$$

• If $c > 0$ and then either both series converge or diverge.
• If $c = 0$ and $\sum b_n$ converges then $\sum a_n$ converges

### Ratio test #

$$\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \rho$$

• If $\rho < 1$, the series converges
• If $\rho > 1$, the series diverges
• If $\rho = 1$, the test is inconclusive

## Taylor series #

Approximation of graphs by polynomials Why? For making computations / approximations faster. More powers makes the approximation more accurate.