Show that the series diverges....

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Geometric Series

A geometric series is a series where each term is obtained from the previous term by multiplying it by a constant factor known as the common ratio. The general form is a + ar + ar² + ar³ + …, where 'a' is the first term and 'r' is the common ratio. This concept is central to understanding the behavior of series that follow a multiplicative pattern.

Common Ratio and Convergence Criteria

For a geometric series, the convergence or divergence of the series is determined by the absolute value of the common ratio. Specifically, a geometric series converges if and only if |r| < 1, because the terms decrease in magnitude sufficiently fast to yield a finite sum; if |r| ? 1, the terms do not diminish and the series diverges. This criterion is crucial in analyzing series of this form, as seen in the provided example where the common ratio exceeds 1.

Divergence Test

The divergence (or nth term) test stipulates that if the limit of the sequence’s terms does not approach zero, then the series must diverge. In the context of a geometric series with a common ratio greater than 1, the terms grow larger and cannot approach zero, thereby violating one of the necessary conditions for convergence. This test provides a straightforward method to identify divergence in many infinite series.

Transcript

Please give Ace some feedback