Series and Convergence

Introduction to Series

A series is the sum of the terms of a sequence. In Calculus II, understanding the convergence or divergence of series is essential for analyzing infinite sums and their properties.

• Convergent series: A series whose sum approaches a finite value as the number of terms increases.

• Divergent series: A series whose sum does not approach a finite value.

• Partial sum: The sum of the first n terms of a series.

Partial Fractions and Telescoping Series

Telescoping Series

Some series can be simplified using partial fraction decomposition, which often leads to a telescoping sum. In a telescoping series, many terms cancel out, making it easier to find the sum.

• Partial fraction decomposition: Expressing a rational function as a sum of simpler fractions.

• Telescoping sum: A series where consecutive terms cancel, leaving only a few terms after summation.

Examples from your homework:

Given , decompose the denominator:

1. Factor:

2. Decompose:

3. Solve for A and B, then sum the series and check for convergence.

Geometric Series

Definition and Convergence

A geometric series is a series of the form , where a is the first term and r is the common ratio.

• Converges if

• Diverges if

• Sum formula: for

Example:

is a geometric series with , .

Converting Repeating Decimals to Rational Numbers

Repeating Decimals

Any repeating decimal can be expressed as a rational number (fraction).

• Let

• Multiply both sides by 100:

• Subtract:

• Solve:

Testing Series for Convergence

Common Tests

Several tests are used to determine if a series converges or diverges:

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

• Limit Comparison Test: Take the limit of the ratio of terms.

• Root Test: Use .

• Ratio Test: Use .

• p-Series Test: converges if .

Example:

is a p-series with , so it diverges.

Special Series Forms

Series with Rational Expressions

Some series involve rational expressions that can be simplified or tested for convergence using algebraic manipulation.

• For , consider the behavior as .

• If the numerator and denominator grow at the same rate, the series may diverge.

Series with Complex Denominators

For series like , use comparison or limit comparison tests to determine convergence.

Summary Table: Series Types and Convergence

Practice Problems (from Homework)

• Determine convergence/divergence and find sums for the following series:

• (partial fractions, telescoping)

• (geometric series)

• (geometric series)

• Series: (identify type and sum)

• Convert to a rational number

• (test for convergence)

• (p-series)

• (comparison test)

Additional info: These problems cover key concepts in Chapter 11: Sequences and Series, including geometric series, telescoping series, p-series, and rational expressions. Understanding these types and their convergence criteria is essential for success in Calculus II.