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

**Question 1:** Investigate **(pointwise) convergence** and
**absolute convergence** of the given series of functions.

**Solution:** **Absolute convergence:**
Consider a general *x* from the intersection of domains, take it as a
fixed parameter and test convergence of the series of real numbers
*f*_{k}(*x*)|.*x*. The region of absolute convergence is the set of all
*x* for which the series with absolute values converges.

**Convergence:** Consider a general *x* from the intersection of domains,
take it as a fixed parameter and test convergence of the series of real
numbers *f*_{k}(*x*).*x*. The region of convergence is the set of all *x* for
which this series converges. Note that now the terms need not be
non-negative, often the resulting series is not even alternating, so the
region of convergence is usually rather harder to find compared to the
region of absolute convergence.

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

**Solution:**
First choose some reasonable candidate for the set of uniform convergence
*M*, namely some subset of the region of convergence.
In a typical case you do not get uniform convergence on the
whole region of convergence, but you can get it if you take away small
neighborhoods of endpoints of this region of convergence, often you also have
to cut away unbounded parts. Very often there is uniform convergence on any
bounded closed interval that is a subset of the region of convergence.

Having chosen one such likely set *M*, how do you determine uniform
convergence? In very rare cases the sum *f* of the series is known.
Then investigate

If *M*_{N}*N* goes to
infinity, then the uniform convergence on *M* is proved.

However, in pretty much all cases the sum *f* is not known. Then we
most often use the
Weierstrass theorem.

Proceed as follows.

**Step 2.** Investigate the series
*a*_{k}*f*_{k}*M*.

Remark: Sometimes one can take for *a*_{k} numbers
larger than the suprema in order to make investigation of the resulting
series easier, see the way the Weierstrass test was stated. Note also that
investigating the suprema of *f*_{k} can help in
guessing the right *M*. When we investigate suprema over the
whole region of convergence and the resulting *a*_{k}
form a divergent series, it often helps to ask this question: Which part of
the region caused these *a*_{k} to be too large?

The Weierstrass test is rather powerful and it is usually the method of choice, but it is not all-powerful. Then it helps to know some alternatives, for instance the Dirichlet test or the Abel test, see Series of functions in Theory - Series of functions.

**Example:** Investigate convergence of

Since the number *k*^{x} is positive for any real
*x* and any positive integer *k*, convergence and absolute
convergence coincide. If *x* is a fixed number, then we need not look
for any test, this is then a typical
*p*-series*p* = *x*)*x* > 1.

Conclusion: The region of (absolute) convergence is the interval

We know that convergence of
*p*-series*p* (in our setting with growing *x*), but
it gets worse as we get near 1 with our parameter. Thus we suspect that the
region around 1 will spoil uniform convergence. Since we do not know the sum
of the given series (only for some special values of *x*), we resort to
the Weierstrass test. First we will see whether we could get uniform
convergence on the whole region of convergence
*M* = (1,∞).

The series
*a*_{k}

We try it again, this time with a smaller set *M*. We decide to test
uniform convergence for the interval
*M* = [*a*,∞)*a* > 1.

Then

and since *a* is chosen to be greater than 1, by the
*p*-test this series
converges. Thus Weierstrass's assumptions are satisfied and uniform
convergence follows.

Conclusion: The given series converges uniformly on sets
*M* = [*a*,∞)*a* > 1.

For other examples see Solved Problems - Series of functions.

Manipulating (power) series

Back to Methods Survey - Series
of functions