Convergence and Divergence of Series: Techniques and Applications

Sequences and Infinite Series

Convergence and Divergence of Series

In calculus, understanding whether an infinite series converges or diverges is essential for analyzing functions, approximations, and mathematical models. Several tests and techniques are used to determine the behavior of series.

• Convergent Series: A series $\sum a_n$ converges if the sequence of partial sums approaches a finite limit as $n \to \infty$.

• Divergent Series: A series diverges if the sequence of partial sums does not approach a finite limit.

• Common Series Types: Geometric, p-series, alternating, and telescoping series.

Tests for Convergence and Divergence

Several tests are used to determine the convergence or divergence of a series:

• p-Series Test: The series $\sum \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \leq 1$.

• Geometric Series Test: The series $\sum ar^n$ converges if $|r| < 1$ and diverges if $|r| \geq 1$.

• Comparison Test: Compare with a known convergent or divergent series.

• Limit Comparison Test: If $\lim_{n \to \infty} \frac{a_n}{b_n} = c$ (where $c$ is finite and positive), then $\sum a_n$ and $\sum b_n$ both converge or both diverge.

• Integral Test: If $f(n) = a_n$ is positive, continuous, and decreasing, then $\sum a_n$ converges if $\int_1^\infty f(x) dx$ converges.

• Ratio Test: For $\sum a_n$, if $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L$, then the series converges if $L < 1$, diverges if $L > 1$, and is inconclusive if $L = 1$.

• Root Test: For $\sum a_n$, if $\lim_{n \to \infty} \sqrt[n]{|a_n|} = L$, then the series converges if $L < 1$, diverges if $L > 1$, and is inconclusive if $L = 1$.

Finding the Sum of a Series

Some series can be summed exactly, especially geometric and telescoping series.

• Geometric Series: $\sum_{n=0}^\infty ar^n = \frac{a}{1 - r}$ for $|r| < 1$.

• Telescoping Series: Series where terms cancel out, leaving a finite sum.

Examples and Applications

• Example 1: $\sum_{n=1}^\infty \frac{1}{n^{1.1}}$ is a p-series with $p = 1.1 > 1$, so it converges.

• Example 2: $\sum_{n=1}^\infty n^2 e^{-n}$ converges by the comparison test with a rapidly decreasing exponential.

• Example 3: $\sum_{n=1}^\infty \frac{2n+3}{\sqrt{5n^2+2n+6}}$ diverges by the limit comparison test with $\frac{n}{n}$.

• Example 4: $\sum_{n=1}^\infty \frac{10n}{n^2+3}$ converges by the comparison test with $\frac{1}{n}$.

• Example 5: $\sum_{n=1}^\infty \left(\sqrt{n+1} - \sqrt{n}\right)$ diverges because the terms do not approach zero.

• Example 6: $\sum_{n=1}^\infty \frac{3}{2^n}$ is a geometric series with $r = \frac{1}{2}$, sum is $3$.

• Example 7: $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{6^n}$ is an alternating geometric series, sum is $\frac{1}{5}$.

Summary Table: Series Tests and Their Applications

Test Name	Formula/Condition	Converges When	Diverges When
p-Series Test	$\sum \frac{1}{n^p}$	$p > 1$	$p \leq 1$
Geometric Series	$\sum ar^n$	$|r| < 1$	$|r| \geq 1$
Comparison Test	Compare $a_n$ to $b_n$	If $b_n$ converges and $a_n \leq b_n$	If $b_n$ diverges and $a_n \geq b_n$
Limit Comparison Test	$\lim_{n \to \infty} \frac{a_n}{b_n} = c$	$c$ finite and $b_n$ converges	$c$ finite and $b_n$ diverges
Integral Test	$\int_1^\infty f(x) dx$	Integral converges	Integral diverges
Ratio Test	$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$	$L < 1$	$L > 1$
Root Test	$L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$	$L < 1$	$L > 1$

Additional info:

• These questions cover material from Chapter 10: Sequences and Infinite Series, including convergence tests and sum calculations.

• Understanding these tests is essential for further study in calculus, especially in power series and differential equations.