Code Scholars Tutoring · AP Calculus BC

The nth-term divergence test and its limits

Unit 10 · Topics 10.3 · Detailed notes, a worked example, and a self-check with reasoning.

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Understand the idea

If the terms of a series do not approach zero, the series cannot converge. This follows from the partial-sum definition and provides a quick way to rule out convergence. It is a divergence test, not a general test that proves convergence.

Compute the limit of the entire term, including signs or oscillation. A nonzero limit, an infinite limit, or failure to have a limit rules out convergence. Do not confuse the term's limit with a ratio of successive terms; those are different tests.

When the term limit is zero, the test gives no conclusion. Both convergent and divergent series can have this behavior. The next step is to use structure such as a geometric pattern, a p-series, comparison, alternating signs, or factorial growth.

\sum a_n\text{ convergent}\ \Longrightarrow\ a_n\to0;\qquad a_n\nrightarrow0\ \Longrightarrow\ \sum a_n\text{ divergent}

A method you can apply

Identify the exact term being summed.
Evaluate its limit as the index increases without bound.
If the limit is zero, explicitly call the test inconclusive and choose another method.

Worked example

Determine whether $\displaystyle \sum_{n=1}^{\infty}(4n+1)/(2n+3)$ converges.

Divide numerator and denominator of the term by $\displaystyle n$ .
The term limit is $\displaystyle \lim(4+1/n)/(2+3/n)=2$ .
The terms therefore do not approach zero.

The series diverges by the nth-term divergence test.

Try it yourself

What does the nth-term test tell you about $\displaystyle \sum_{n=1}^{\infty}1/\sqrt n$ ?

Show solution and reasoning

The term limit is zero, so this test is inconclusive. A separate p-series argument, with $\displaystyle p=1/2$ , establishes divergence; the zero term limit does not establish convergence.