The Comparison Test

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The Limit Comparison Test

There are two extensions of the basic comparison test:

Theorem: Suppose that $\displaystyle{\sum_{n=1}^\infty a_n}$ and $\displaystyle{\sum_{n=1}^\infty b_n}$ are positive series; that $c$ is a positive constant; and that $N$ is some positive integer.

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

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

Limit Comparison Test: Let $\displaystyle{\sum_{n=1}^\infty a_n}$ and $\displaystyle{\sum_{n=1}^\infty b_n}$ be positive series. If $$\displaystyle{\lim_{n \to \infty} \frac{a_n}{b_n}}=c,$$ for some positive number $c$, then either both series converge or both diverge.

Example: To determine whether the series $\displaystyle\sum_{n=1}^\infty \frac{4^n}{2^n+3^n}$ converges or diverges, we'll look for a series that "behaves like" it when $n$ is large. Since $$ \frac{4^n}{2^n+3^n}\approx \frac{4^n}{3^n}, $$ when $n$ is large, we'll use $\displaystyle\sum_{n=1}^\infty\left(\frac43\right)^n$ for comparison. Since $$ \lim_{n\to\infty}\frac{ \frac{4^n}{2^n+3^n}}{\frac{4^n}{3^n}}= \lim_{n\to\infty}\frac{ 3^n}{2^n+3^n}= \lim_{n\to\infty}\frac{ 3^n}{3^n\left(\frac{2^n}{3^n}+1\right)}= 1>0, $$ and the geometric series $\displaystyle\sum_{n=1}^\infty\left(\frac43\right)^n$ diverges, we can conclude that our original series diverges as well.

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Differential Equations

Models of Growth

Infinite Sequences

Infinite Series

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 Limit Comparison Test