Convergence of series of functions: Methods survey

We are given a series of functions ∑ f_k(x) and we assume that the intersection of all their domains is not empty. Usually we ask about two things: What is the region of (absolute) convergence and whether the series converges uniformly at least somewhere.

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)|. Note that terms of this series are all non-negative, so one can use all the wonderful tests. The answer (convergent, divergent) in most cases depends on the value of the parameter 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). The answer (convergent, divergent) in most cases depends on the value of the parameter 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 tend to zero as 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.

Step 2. Investigate the series ∑ a_k (of real non-negative numbers). If this series converges, then the series of functions ∑ f_k converges uniformly on 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 (here p = x) whose convergence we should remember. Thus we know that this series converges (and absolutely converges) exactly if x > 1.
Conclusion: The region of (absolute) convergence is the interval (1,∞).

We know that convergence of p-series improves with growing 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,∞). We get

The series ∑ a_k that we obtain in this way is the famous harmonic series that we know to be divergent. Thus the Weierstrass test failed, which we actually expected.

We try it again, this time with a smaller set M. We decide to test uniform convergence for the interval M = [a,∞) for some arbitrary fixed a > 1. Now we have

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,∞) for any a > 1.

For other examples see Solved Problems - Series of functions.

Manipulating (power) series
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