Calculus 2 Lecture Notes, Section 8.3

Calc 2 Lecture NotesSection 8.3Page 1 of 5

Section 8.3: The Integral Test and Comparison Tests

Big idea:. We can determine whether some series diverge or converge by comparing the series to another series with known convergence/divergence, or by thinking of the series as a Riemann sum, and comparing the area to known integrals that converge/diverge.

Big skill:. You should be able to determine the convergence or divergence of certain series using the integral and comparison tests.

Consider the (divergent) harmonic series . To help us visualize why this series diverges, we can compare it to a Riemann sum for , where we use left endpoint evaluation and a rectangle width of . Clearly, from the picture below, the area of the Riemann sums is greater than the area under the curve. So, we can write . Since the integral diverges… , the sum must diverge as well. Thus, diverges.

Now consider the (convergent) series . To help us visualize why this series converges, we can compare it to a Riemann sum for , where we use right endpoint evaluation and a rectangle width of . Clearly, from the picture below, the area of the Riemann sums is less than the area under the curve.

So, we can write . Since the integral converges… , we can see that the original sum converges using a little algebra:

Theorem 3.1: The Integral Test for Convergence of a Series

If f(k) = ak for all k = 1, 2, 3, …, and f is both continuous and decreasing, and f(x)  0 for x 1, then and either both converge or both diverge.

Practice: determine whether converges or diverges.

Corollary: p-Series

The p-Series converges if p > 1 and diverges if p 1.

Practice: Prove the p-Series corollary.

Some known p-Series Values:

Theorem 3.2: Error Estimate for the Integral Test

Suppose that f(k) = ak for all k = 1, 2, 3, …, where f is both continuous and decreasing, and
f(x)  0 for x 1, and that converges. Then, the remainder Rn satisfies .

Practice:

The 1000th partial sum of the series is S1000 1.6439345667. What error is associated with that partial sum?

If you want to compute a partial sum of such that your answer is within 0.01 of the exact answer, how many terms must you include in your sum?

Theorem 3.3: The Comparison Test for Convergence of a Series

Suppose that 0 akbk for all k .

If converges, then converges, too.
If diverges, then diverges, too.

Practice:

Determine whether converges or diverges.

Theorem 3.4: The Limit Comparison Test for Convergence of a Series

Suppose that ak, bk > 0, and that for some finite number L, . Then, either and both converge or both diverge.

Practice:

Determine whether converges or diverges.