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 $f_1,f_2,\dots ,f_n\dots$ 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 $li{m}_{n\to \infty }\sqrt{{\int }_a^b|f_n(x)-f(x){|}^{2}dx}=0$ tldr; the “average error” goes to 0.

• the sequence pointwise converges to $f$, if for any $x\in [a,b]$ ${\lim }_{n\to \infty }(f_n(x)-f(x))=0$ tldr; the error at each point goes to 0.

• The sequence uniformly converges to $f$ if $li{m}_{n\to \infty }{\max }_{[a,b]}|f_n(x)-f(x)|=0$ 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.

Related

• Fourier Series