# 2. Sequences and series of functions

### Sequences of functions, pointwise convergence

A sequence of functions is a sequence with functions as elements , defined on a set . E.g.: .

We shall say that a sequence of functions converges pointwise to a function (), if for all we have (as a sequence of reals).

E.g.: converges pointwise to na , because for all we have .

### Uniform convergence

A sequence converges uniformly on to (), if:

Meaning that in every point it converges at least equally fast. In other words, a sequence of functions converges uniformly if the sequence converges to zero.

E.g., sequence converges uniformly to to , because, if , then let and then for any :

because .

The same sequence does not converge uniformly on the whole set , since if, e.g. , for any , I can find , such that . Indeed, let . Then .

The following important theorem holds. If all are continuous on a set and , then is also continuous.

The above may not hold in the case of pointwise convergence. E.g. converges pointwise on to

which is not continuous. Obviously, is not uniformly convergent on .

### Series of functions

Similarly as in the case of series of reals we can create a series of functions out of a sequence of functions. A sum of such a sequence is as before the limit of sequence of partial sums. Therefore we can study sets of arguments on which a given series is convergent. E.g. series converges for any to .

### Weierstrass criterion

This criterion seems quite clear: if there exists , such that for any and any the following inequality holds: , where is convergent, then is convergent on absolutely and uniformly.

E.g. converges on the whole , because and is convergent.

### Power series

A series of functions of form is called a power series. E.g. , is convergent for .