Problem: Investigate convergence of the following series of functions:

Solution: First we investigate pointwise convergence. We treat x as a parameter and test convergence of the resulting series of real numbers. Since cosine hyperbolic is always positive, the given series has non-negative terms and we can apply suitable tests to work out its convergence (that is also absolute convergence).

What tests seem good here? Taking the k-th root of the cosine hyperbolic does not look too inviting, so how about the Ratio test?

Thus the test was inconclusive. What other test can we use? There is only one reasonable choice, limit comparison. We see that for large k, the numerator is essentially equal to cosh(x), which is a constant (we take x as a parameter now) and so it can be taken out of the series. We thus guess that our series behaves like the series with 1/k2. We have to justify this guess.

Since the series on the right converges (see the p-test), also the series on the left converges. Note that this convergence does not depend on x.

Conclusion: The given series converges (absolutely) on the whole real line (which is the region of (absolute) convergence).

How about uniform convergence? Since we do not actually know the sum of the given series, we cannot use direct methods. However, the comparison above suggests that the Weierstrass test should work fine on bounded sets. Indeed, fix any A < B and consider the set M=[A,B]. Using the fact that cosine hyperbolic is even and increasing on (0,∞), for x from M we can estimate

Therefore

Since the series ∑ ak converges (see above), the given function series converges uniformly on M.

Conclusion: The given series converges uniformly on any bounded closed interval.


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