The Comparison Test

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The Basic Comparison Test

Theorem: If $\displaystyle{\sum_{n=1}^\infty a_n}$ and $\displaystyle{\sum_{n=1}^\infty b_n}$ are series with non-negative terms, then:

If $\displaystyle{\sum_{n=1}^\infty b_n}$ converges and $a_n \le b_n$ for all $n$, then $\displaystyle{\sum_{n=1}^\infty a_n}$ converges.

If $\displaystyle{\sum_{n=1}^\infty b_n}$ diverges and $a_n \ge b_n$ for all $n$, then $\displaystyle{\sum_{n=1}^\infty a_n}$ diverges.

In fact, 1. will work if $a_n\le b_n$ for all $n$ larger than some finite positive $N$, and similarly for 2.

Example 1: The series $\displaystyle \sum_{n=1}^\infty\frac{2^n}{3^n+1}$ converges, since $$ \frac{2^n}{3^n+1}\le \frac{2^n}{3^n} $$ and we know that the geometric series $\displaystyle \sum_{n=1}^\infty\left(\frac{2}{3}\right)^n$ is a convergent geometric series, with $r=\frac23<1$.

The video explains the test, and looks at an example.

Example 2: Test the series $\displaystyle\sum_{k=1}^\infty\frac{\ln k}{k}$ for convergence or divergence.

DO: Do you think this series converges? Try to figure out what to compare this series to before reading the solution

Solution 2: $\displaystyle\frac{\ln k}{k}\ge\frac1k$, and the harmonic series with terms $\frac1k$ diverges, so our series diverges.

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Example 3: Test the series $\displaystyle\sum_{n=1}^\infty\frac{1}{5n+10}$ for convergence or divergence. DO: Try this before reading further.

Solution 3: The terms look much like the harmonic series, and when we compare terms, we see that $\displaystyle\frac{1}{5n+10}\le\frac1n$. But the harmonic series diverges. Our terms are smaller than those of a divergent series, so we know nothing. Let's compare to $\displaystyle\frac1{n^2}$. The series $\displaystyle\sum\frac{1}{n^2}$ is a convergent $p$-series, with $p=2$. But when we compare terms, we get $\displaystyle\frac{1}{5n+10}\ge\frac1{n^2}$ as long as $n\ge7$, so our terms are larger than those of a convergent series, and this comparison also tells us nothing. We will use the limit comparison test (coming up) to test this series.

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

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Partial Fractions

Strategies of Integration

Improper Integrals

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Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Partial Derivatives

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

The Basic Comparison Test