# Infinite sequences and limits

## More on sequences

### Limit of a sequence

A sequence converges to a limit if

Can converge to a number (1/x)

Can converge to +/- infinity (x)

Otherwise, does not converge (1,-1,1,-1…)

Superior and inferior limits

A bounded increasing sequence converges to least upper bound

#### Identifying the limit of a sequence

Direct comparison test

Root test

## Divergent series

### Partial sum

Take a series. We can define the partial sum as:

$$s_k=\sum_{i=1}^ka_i$$

### Cesàro sum

The Cesàro sum is the limit of the average of the first $$n$$ partial sums.

That is:

$$\lim_{n\rightarrow \infty }\dfrac{1}{n}\sum_{k=1}^ns_k$$

Consider the sequence $$\{1,-1,1,-1,...\}$$

The partial sum is:

$$s_k=\sum_{i=1}^ka_i$$

$$s_k=k\mod(2)$$

The Cesàro sum is: $$\lim_{n\rightarrow \infty }\dfrac{1}{n}\sum_{k=1}^ns_k$$

$$\lim_{n\rightarrow \infty }\dfrac{1}{n}\sum_{k=1}^nk\mod(2)$$

$$\dfrac{1}{2}$$