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 .
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 :
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 .
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.
A series of functions of form is called a power series. E.g. , is convergent for .
Radius of convergence
Given a series , a real number equal to supremum of the set of arguments for which this series is convergent is called the radius of convergence of this series. Actually, the radius of convergence up to two points describes the convergence of a power series, because the following fact.
If is the radius of convergence of then this series is absolutely convergent on and not convergent on . The theorem does not describe convergence for . Therefore to describe the set of convergence of a given power series, it suffices to calculate the radius of convergence and additionally check what happens for .
Calculating the radius of convergence
Given series we can deduce from d’Alembert and Cauchy criterion the radius of convergence. The following two facts hold:
- if , then if , then , but if , then ,
- if , then if , then , but if , then .
E.g. for we have , so .