Feedback

What do you think about us?

Your name

Your email

Message

The nth Term Test for Divergence is a crucial tool in calculus for determining the divergence of infinite series. It states that if the limit of the sequence of terms does not approach zero as n approaches infinity, the series diverges. This test is vital for initial series analysis but cannot confirm convergence. Examples like the Harmonic series and P-series illustrate its application, while the Integral Test offers a more comprehensive analysis for both convergence and divergence.

Show More

## The \(n^{th}\) Term Test for Divergence

### Definition

The \(n^{th}\) Term Test for Divergence is a method used to determine if a series diverges by examining the limit of the sequence of terms

### Misconceptions

Belief that approaching zero guarantees convergence

A common misconception is that if the terms of a series approach zero, the series must converge, but this is not always the case

Limit of sequence must be zero for convergence

Another misconception is that the limit of the sequence of terms must be zero for a series to converge, but this is only a necessary condition, not a sufficient one

### Rationale

The \(n^{th}\) Term Test for Divergence is based on the contrapositive that if a series converges, the limit of the sequence of terms must be zero

## The Integral Test

### Definition

The Integral Test is a method used to determine the convergence or divergence of a series by comparing it to an improper integral

### Applicability

The Integral Test can be applied to series with terms from a positive, continuous, and monotonically decreasing function

### Comprehensive tool

Unlike the \(n^{th}\) Term Test for Divergence, the Integral Test can confirm both convergence and divergence of a series