Integral and Comparison Tests for Series Convergence

Section 10.3: The Integral Test

Introduction to the Integral Test

The Integral Test is a method used to determine the convergence or divergence of infinite series whose terms are positive and decrease monotonically. It is particularly useful when the series terms resemble a function that is easy to integrate.

• Key Idea: Compare the sum of a series to the value of an improper integral.

• Typical Series: $\sum\limits_{n=1}^{\infty} \frac{1}{n^p}$ (p-series)

Theorem: The Integral Test

If $a_n = f(n)$, where $f$ is a continuous, positive, decreasing function for $x \geq N$ (for some integer $N$), then the series $\sum\limits_{n=N}^{\infty} a_n$ and the improper integral $\int_N^{\infty} f(x)\,dx$ both converge or both diverge.

• Convergence: If the integral converges, so does the series.

• Divergence: If the integral diverges, so does the series.

Application to p-Series

The p-series $\sum\limits_{n=1}^{\infty} \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \leq 1$.

• For $p = 2$, the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^2}$ converges.

• For $p = 1$, the harmonic series $\sum\limits_{n=1}^{\infty} \frac{1}{n}$ diverges.

Examples and Visualizations

Consider the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^2}$:

• The sum of the areas of rectangles with height $\frac{1}{n^2}$ and width 1 is less than the area under $y = \frac{1}{x^2}$ for $x \geq 1$.

• The improper integral $\int_1^{\infty} \frac{1}{x^2} dx = 1$ is convergent, so the series converges.

For the harmonic series $\sum\limits_{n=1}^{\infty} \frac{1}{n}$, the corresponding integral $\int_1^{\infty} \frac{1}{x} dx$ diverges, so the series diverges.

Estimating Sums and Remainders

The Integral Test also provides a way to estimate the error (remainder) when approximating the sum of a convergent series by its first $n$ terms.

• The remainder $R_n = S - s_n$ satisfies:

$\int_{n+1}^{\infty} f(x) dx \leq R_n \leq \int_n^{\infty} f(x) dx$

Section 10.4: Comparison Tests

Direct Comparison Test

The Direct Comparison Test is used to determine the convergence or divergence of a series by comparing it to another series with known behavior.

• If $0 \leq a_n \leq b_n$ for all $n$ and $\sum b_n$ converges, then $\sum a_n$ also converges.

• If $0 \leq b_n \leq a_n$ for all $n$ and $\sum b_n$ diverges, then $\sum a_n$ also diverges.

Limit Comparison Test

The Limit Comparison Test is useful when the terms of two series are similar for large $n$ but not necessarily ordered.

• If $\lim\limits_{n \to \infty} \frac{a_n}{b_n} = c$ where $0 < c < \infty$, then $\sum a_n$ and $\sum b_n$ both converge or both diverge.

• If the limit is 0 and $\sum b_n$ converges, then $\sum a_n$ converges.

• If the limit is $\infty$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

Estimating Sums Using Comparison

When using the Comparison Test, the remainder of the series being tested can be estimated by the remainder of the comparison series, especially if the comparison series is a p-series or geometric series.

Summary Table: Partial Sums of a Divergent Series

The following table shows the partial sums $s_n$ for the series $\sum\limits_{i=1}^n \frac{1}{\sqrt{i}}$, illustrating divergence as $n$ increases:

n	$s_n = \sum\limits_{i=1}^n \frac{1}{\sqrt{i}}$
5	3.2317
10	5.0210
50	12.7524
100	18.5896
500	43.2834
1000	61.8010
5000	139.9681

Key Formulas and Theorems

• Integral Test: $\sum\limits_{n=N}^{\infty} a_n$ converges if and only if $\int_N^{\infty} f(x) dx$ converges.

• Direct Comparison Test: Compare $a_n$ to $b_n$ termwise.

• Limit Comparison Test: $\lim\limits_{n \to \infty} \frac{a_n}{b_n} = c$ (with $0 < c < \infty$) implies both series behave the same.

• Remainder Estimate (Integral Test): $\int_{n+1}^{\infty} f(x) dx \leq R_n \leq \int_n^{\infty} f(x) dx$

Examples

• Example 1: $\sum\limits_{n=1}^{\infty} \frac{1}{n^2}$ converges by the Integral Test.

• Example 2: $\sum\limits_{n=1}^{\infty} \frac{1}{n}$ diverges (harmonic series).

• Example 3: $\sum\limits_{n=1}^{\infty} \frac{1}{n^{1/2}}$ diverges (p-series with $p = 1/2 < 1$).

• Example 4: $\sum\limits_{n=1}^{\infty} \frac{\ln n}{n}$ diverges by the Integral Test.

Additional info: The notes above include all major results, visualizations, and remainder estimates for the Integral Test and both Comparison Tests, as well as a summary table for divergent series. All images included are directly relevant to the explanations provided.