Convergence and Divergence of Series and Sequences; Power Series and Taylor Series

Sequences and Series

Introduction to Sequences and Series

Sequences and series are fundamental concepts in calculus, especially in the study of infinite processes. A sequence is an ordered list of numbers, while a series is the sum of the terms of a sequence. Understanding their convergence or divergence is essential for advanced calculus topics.

Convergence and Divergence of Series

• Convergent Series: A series $\sum_{n=1}^{\infty} a_n$ converges if the sequence of partial sums approaches a finite limit as $n \to \infty$.

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

Common Tests for Convergence

• n-th Term Test: If $\lim_{n \to \infty} a_n \neq 0$, the series diverges.

• Geometric Series Test: A geometric series $\sum ar^n$ converges if $|r| < 1$.

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

• 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 $0 < c < \infty$), then both series behave the same.

• Ratio Test: For $\sum a_n$, compute $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$:

  • If $L < 1$, the series converges absolutely.

  • If $L > 1$ or $L = \infty$, the series diverges.

  • If $L = 1$, the test is inconclusive.

• Root Test: For $\sum a_n$, compute $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$ (similar conclusions as the Ratio Test).

• Alternating Series Test (Leibniz Test): For $\sum (-1)^n a_n$ with $a_n > 0$:

  • If $a_n$ is decreasing and $\lim_{n \to \infty} a_n = 0$, the series converges.

Absolute and Conditional Convergence

• Absolutely Convergent: If $\sum |a_n|$ converges, then $\sum a_n$ converges absolutely.

• Conditionally Convergent: If $\sum a_n$ converges but $\sum |a_n|$ diverges, the series is conditionally convergent.

Examples

• Geometric Series: $\sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n$ converges because $|r| = \frac{1}{2} < 1$.

• p-Series: $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges since $p = 2 > 1$.

Strategies for Series Convergence

• Identify the type of series (geometric, p-series, alternating, etc.).

• Apply the appropriate test(s) based on the form of the series.

• For complicated series, consider the Ratio or Root Test.

• For alternating series, check for absolute or conditional convergence.

Sequences: Convergence, Monotonicity, and Boundedness

Convergence of Sequences

• A sequence $a_n$ converges to $L$ if $\lim_{n \to \infty} a_n = L$.

• If the limit does not exist or is infinite, the sequence diverges.

Monotonicity and Boundedness

• Increasing Sequence: $a_{n+1} > a_n$ for all $n$.

• Decreasing Sequence: $a_{n+1} < a_n$ for all $n$.

• Monotonic Sequence: Sequence is either increasing or decreasing.

• Bounded Above: There exists $M$ such that $a_n \leq M$ for all $n$.

• Bounded Below: There exists $m$ such that $a_n \geq m$ for all $n$.

• Bounded Sequence: Sequence is both bounded above and below.

Examples

• $a_n = \frac{1}{n}$ is decreasing, bounded below by 0, and converges to 0.

• $a_n = (-1)^n$ is not monotonic and does not converge.

Power Series and Radius of Convergence

Power Series

A power series is an infinite series of the form $\sum_{n=0}^{\infty} c_n (x - a)^n$, where $c_n$ are coefficients and $a$ is the center.

Radius and Interval of Convergence

• The radius of convergence $R$ is the distance from $a$ within which the series converges.

• Find $R$ using the Ratio Test:

  • $R = \lim_{n \to \infty} \left| \frac{c_n}{c_{n+1}} \right|$

• The interval of convergence is $(a - R, a + R)$, possibly including endpoints (test separately).

Examples

• $\sum_{n=0}^{\infty} \frac{(x-2)^n}{n!}$ converges for all $x$ (radius is infinite).

Taylor Series

Definition and Construction

The Taylor series for a function $f(x)$ about $x = a$ is:

$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$

• Maclaurin Series: Special case where $a = 0$.

Common Taylor Series Expansions

• $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$

• $\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$

• $\ln x$ about $x=1$: $\ln x = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} (x-1)^n$

Applications

• Approximating functions near a point.

• Solving differential equations.

• Representing functions as infinite polynomials.

Summary Table: Series Convergence Tests

Test	When to Use	Convergence Criteria
n-th Term Test	Any series	If $\lim a_n \neq 0$, diverges
Geometric Series	$ar^n$	$|r| < 1$: converges
p-Series	$1/n^p$	$p > 1$: converges
Comparison Test	Similar to known series	Compare with convergent/divergent series
Ratio Test	Factorials, exponentials	$L < 1$: converges
Root Test	Powers of n	$L < 1$: converges
Alternating Series	$(-1)^n a_n$	$a_n$ decreasing, $\to 0$

Practice Problems Overview

• Determine convergence/divergence for various series using appropriate tests.

• Analyze sequences for monotonicity and boundedness.

• Find the radius and interval of convergence for power series.

• Find Taylor series for given functions about specified points.

Additional info: The problems provided cover topics from Chapter 11 (Sequences and Series), including convergence tests, power series, and Taylor series, which are central to Calculus II.