Lecture on Convergence of Series

This set of flashcards reviews key concepts regarding the convergence of numerical series, including definitions and tests applicable to determine their behavior.

What does it mean for a series to be absolutely convergent?

A series is absolutely convergent if the series of absolute values of its terms converges.

What is the Ratio Test used for in convergence?

The Ratio Test is a method to determine the convergence or divergence of infinite series.

How can you identify a geometric series?

A geometric series is identified by having a constant ratio between consecutive terms.

What is the significance of alternating series in convergence?

An alternating series may converge even if the series of absolute values does not converge.

What is the relationship between convergent and divergent series?

If a series converges, it may also imply conditions under which related series may diverge.

What notation is commonly used to represent the limit in convergence series?

The limit can be represented as 'lim' followed by the terms of the series.

What is a practice problem for testing absolute convergence?

Determine if the series ∑ (1/n^2) is absolutely convergent.

What is a practice problem for the Ratio Test?

Use the Ratio Test to analyze the convergence of the series ∑ (n! / 2^n).

Provide a practice problem for identifying a geometric series.

Identify if the series ∑ (3^n) is a geometric series and state its common ratio.

Create a practice problem regarding alternating series.

Determine if the series ∑ (-1)^n / n converges and explain why.

Formulate a practice problem comparing convergent and divergent series.

If ∑ (1/n) diverges, what can be said about the series ∑ (1/n^2)?

Give a practice problem involving limits in series.

Find the limit of the series as n approaches infinity for ∑ (1/n).