# Convergence

I think there is only a fairly few amount of disciplines where people who are really inquisitive about the world go to.

### Math Convergence

Some in Series.

From MATH213.

Definition 3: Some Types of Convergence

If $f_{1},f_{2},β¦,f_{n}β¦$ is a sequence of $L_{2}$ functions defined on $[a,b]$ then we say that

the sequence converges in the $L_{2}([a,b])$ norm, or converges in the mean or converges almost everywhere, to $f$ if $lim_{nββ}β«_{a}β£f_{n}(x)βf(x)β£_{2}dxβ=0$ tldr; the βaverage errorβ goes to 0.

the sequence

pointwise convergesto $f$, if for any $xβ[a,b]$ $lim_{nββ}(f_{n}(x)βf(x))=0$ tldr; the error at each point goes to 0.The sequence

uniformly convergesto $f$ if $lim_{nββ}max_{[a,b]}β£f_{n}(x)βf(x)β£=0$ If the maximum does not exist then we replace it with the smallest upper bound (called the sup). tldr; the maximum error converges to 0.

What is

`f(x)`

?How is it defined? LIke I get that we have a sequence.