We are given a sequence of functions
*f*_{k}(*x*)}

**Question 1:** Investigate **(pointwise) convergence** of the given
sequence of functions.

**Solution:** Consider a general *x* from the intersection of domains,
take it as a fixed parameter and evaluate the limit of the sequence of real
numbers *f*_{k}(*x*)}*k* going to infinity.

For some *x* this will diverge. For some *x* this limit converges,
call the resulting number
*f* (*x*).*x* for which this limit converges forms the region of
convergence, on it we have the function *f* obtained as above and we
say that the sequence
*f*_{k}}*f* there.

**Question 2:** Investigate **uniform convergence** of the given
sequence of functions.

**Solution:** First, find the limit *f* of
*f*_{k}}

Next, guess a set *M* (a subset of the region of convergence) on which
you suspect uniform convergence. Typically you start with the region of
convergence, or the region of convergence without small neighborhoods of its
endpoints.

For a fixed *k*, evaluate

*M*_{k} = sup{| *f* (*x*) − *f*_{k}(*x*)|,
*x* from *M* }.

If *M*_{k}→0*k* going to infinity, uniform
convergence on *M* is proved.

If not, then most likely the set *M* is too ambitious. Try to guess a
smaller set, analysis of the suprema above may help in identifying which
parts of the original *M* caused troubles.

**Example:** Investigate convergence of the sequence

**Solution:** First we investigate convergence. We consider *x* to
be a parameter and evaluate limit in *k*.

This limit existed for all values of *x*, so the region of convergence
is the whole real line and the given sequence converges there to the
function
*f* (*x*) = *x*^{2}.

Is this convergence uniform? We look at the difference. For a fixed *k*
one gets

Since all suprema are infinity, there is no way they can go to zero and thus
we do not have uniform convergence on the whole real line. Obviously the
problem is that *x* is allowed to become arbitrarily large. Thus we
guess that we have a better chance if we investigate uniform convergence on
a closed interval *M* = [−*a*,*a*]*a*. Now we get

When we send *k* to infinity, then *M*_{k} tends to
zero, which proves that the given sequence
*f*_{k}(*x*)}*f* (*x*) = *x*^{2}*M* = [−*a*,*a*]*a*.

In fact, a similar argument shows a somewhat more general and simpler to
state fact that the convergence is uniform on every bounded subset of real
numbers.

For other examples see Solved Problems - Series of functions.

Convergence of series of functions

Back to Methods Survey - Series
of functions