Convergence of Series: BC Calc

Front

What is a Harmonic Series?

Back

A series of the form Σ 1/n, where n is a positive integer. It diverges as n approaches infinity.

Front

What does it mean for a series to converge?

Back

A series converges if the sum of its terms approaches a finite number as more terms are added.

Front

What is a P-series?

Back

A series of the form Σ 1/n^p, where p is a positive constant. It converges if p > 1 and diverges if p ≤ 1.

Front

What is a Geometric Series?

Back

A series of the form Σ ar^n, where a is the first term and r is the common ratio. It converges if |r| < 1.

Front

What is the factorial notation (n!)?

Back

The product of all positive integers up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

Front

Does the series Σ 1/n! converge or diverge?

Back

Converges. The terms decrease rapidly as n increases.

Front

Does the series Σ 4/√n converge or diverge?

Back

Diverges. It behaves like a p-series with p = 1/2, which diverges.

Front

What is the ratio test?

Back

A method to determine the convergence or divergence of a series by examining the limit of the ratio of consecutive terms.

Front

What is the nth-term test for divergence?

Back

If the limit of the terms of a series does not approach zero, the series diverges.

Front

What is a Taylor Series?

Back

A series that represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point.