Series: Guide to Investigating Convergence

Understanding the Convergence of a Series

A series converges to λ if the limit of the sequence of the partial sums of the series is equal to λ

Example (1)

Warning

The Sum of the series

Questions Check, whether the given series is convergent, and if convergent find its sum

Example (2) Telescoping Series

Warning

Examples of this type of telescoping series A Convergent Telescoping Series

Solutions

Examples of this type of telescoping series A Divergent Telescoping Series

Questions I Check, whether the given series is convergent, and if convergent find its sum

Questions II Show that the following series is a telescoping series, and then determine whether it is convergent

The Integral Test

Example (3)

Warning

Questions Check, whether the given series is convergent.

Algebra of Series Convergence

Questions

Divergence Test

Questions Check, whether the given series is convergent.

Convergence Tests

Convergence Tests for Series of Positive Terms 1. Comparison Test 2. Limit Comparison Test 3. Ratio Test 4. Root Test

The Comparison test

Examples

Example (1)

Solution

Definition Order of Magnitude of a Series

Question

The Limit Comparison test

Examples

Solution

The Ratio test

Examples

The Root test

Examples

Definition Alternating Series

Alternating Series Convergence Test

Example

Definition Absolute and Conditional Convergence

Example (1)

Example (2)

The Ratio test for Absolute Convergence

Examples: Investigate the absolute convergence of the following series

More Examples on the Integral Test

Example (1)

Example (2)

Example (3)