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
Direct comparison test
Root test
Take a series. We can define the partial sum as:
\(s_k=\sum_{i=1}^ka_i\)
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: \(\lim_{n\rightarrow \infty }\dfrac{1}{n}\sum_{k=1}^ns_k\)
\(\lim_{n\rightarrow \infty }\dfrac{1}{n}\sum_{k=1}^nk\mod(2)\)
\(\dfrac{1}{2}\)