Series Exercises

Answer

No, the series doesn't have to converge.

The harmonic series is the standard counterexample. The terms of the harmonic series approach zero, but the harmonic series itself diverges. The correct statement is that if a series converges, its terms must approach zero.

The example was sort of like comparing two improper integrals, but instead of two integrals we had one integral and one collection of rectangles trying to approximate the integral.

Answer

Let . This function is decreasing and non-negative on [1,∞), and the nth term of the series is f(n).

Since

diverges by the integral p-test (since  is less than 1), the series must also diverge.

By the magic of the integral test, we get a p-test for series like we had for improper integrals. The series

converges if p > 1 and diverges otherwise.

The big catch to the integral test is that there are integrals we can't evaluate. If we can't evaluate the integral, the integral test is worthless. Sometimes, the gremlins shake the box so hard that we can't decide whether to open it or not. We need another test to figure out if we should open it or not.

There are also situations where we aren't allowed to use the integral test because the function f doesn't satisfy the hypotheses. In order to use the integral test, the function that describes the terms has to be non-negative and decreasing on [c,∞).

Answer

The function f(x) = e^-x is decreasing and non-negative on [0, ∞) so we're allowed to use the integral test. We can integrate f(x).

Since the integral converges, the series converges also.

Answer

The sign of sin does weird stuff, which means the function f(x) = x sin x is sometimes negative. There is no point c for which f(x) is non-negative on [c,∞).

This means we're not allowed to use the integral test.

Answer

The function

is non-negative and decreasing on [1,∞). From ancient history we know that since 0.5 < 1 the integral

diverges. The integral test tells us that the series

must also diverge.

The shorter version is that we could use the p-test for series to say that since

p = 0.5 < 1

we know by the p-test that the series diverges.

Answer

The function factors:

Sadly, from x = 9 onwards this function is negative.

That means we're not allowed to use the integral test.

Answer

The function

is non-negative and decreasing. Integrating this requires a formula:

The arctan function approaches  as x goes to infinity:

This means the integral works out to

We don't know what this number is, but it's a finite number, and that's all that matters. The improper integral converges, and by the integral test the series does too.