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:


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:



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

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


Abel summation