Page 96 - Calculus for the Clueless, Calc II

The bounds on are

Note 2

In this case, this is not too good an approximation.

We will get a better one if we take the fourth partial sum:

The "error," the estimate on the rest of the terms, is that

This is more accuracy than you will probably ever need!!! Lots of things you cannot even integrate.

Example 18—

The harmonic series diverges.

which goes to infinity as b goes to infinity.

Example 19—

The p 2 series converges.

dx = -1/b + 1. Since-1/b goes to 0 as b goes to infinity, this improper integral converges. So does the p 2.

Example 20—

If p > 1, it converges, and if p < 1, it diverges. Just use the integral test. It's easy.

Test 3

The comparison test. Given where 0 < a k < b k,

1. If converges, so does .

2. If diverges, so does .

Let us talk through part 1. The second part can be shown to be equivalent. The partial sums S n of the
series are uniformly bounded because the first N terms are bounded by their maximum and the rest are bounded
by L + ε. Therefore the partial sums of the series also are bounded, being respectfully smaller than those
of . Moreover, since a k > 0, then the partial sums of the a k form an increasing sequence. Now we have an
increasing bounded sequence that has a limit. Therefore, converges.