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 converges if the sequence of partial sums approaches a finite limit as .

• 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 , the series diverges.

• Geometric Series Test: A geometric series converges if .

• p-Series Test: The series converges if and diverges if .

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

• Limit Comparison Test: If (where ), then both series behave the same.

• Ratio Test: For , compute :

  • If , the series converges absolutely.

  • If or , the series diverges.

  • If , the test is inconclusive.

• Root Test: For , compute (similar conclusions as the Ratio Test).

• Alternating Series Test (Leibniz Test): For with :

  • If is decreasing and , the series converges.

Absolute and Conditional Convergence

• Absolutely Convergent: If converges, then converges absolutely.

• Conditionally Convergent: If converges but diverges, the series is conditionally convergent.

Examples

• Geometric Series: converges because .

• p-Series: converges since .

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 converges to if .

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

Monotonicity and Boundedness

• Increasing Sequence: for all .

• Decreasing Sequence: for all .

• Monotonic Sequence: Sequence is either increasing or decreasing.

• Bounded Above: There exists such that for all .

• Bounded Below: There exists such that for all .

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

Examples

• is decreasing, bounded below by 0, and converges to 0.

• 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 , where are coefficients and is the center.

Radius and Interval of Convergence

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

• Find using the Ratio Test:

• The interval of convergence is , possibly including endpoints (test separately).

Examples

• converges for all (radius is infinite).

Taylor Series

Definition and Construction

The Taylor series for a function about is:

• Maclaurin Series: Special case where .

Common Taylor Series Expansions

• about :

Applications

• Approximating functions near a point.

• Solving differential equations.

• Representing functions as infinite polynomials.

Summary Table: Series Convergence Tests

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.