Unit 10 · Topic 10.3

The nth Term Test for Divergence

If the terms of a series do not approach 0, the series diverges: that is the nth term test.

Practice 3: JustificationPractice 4: Communication and Notation

Part 1 · Hook

Why this matters

Pour a cup of water into a bucket every minute, forever, and the bucket overflows: if the pieces you add do not shrink to nothing, an endless sum grows without bound. That is the nth term test, the first test to try on any series. It can prove a series diverges, but it can never prove one converges.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. For a convergent series, the sum is the limit of

  1. its partial sums SnS_n
  2. its terms ana_n
  3. its first term
  4. the ratio of consecutive terms
Show the answer

By definition, ∑an\sum a_n=limn→∞Sn= \lim_{n \to \infty} S_n.

  • Correct: its partial sums SnS_n:
  • its terms ana_n:
  • its first term:
  • the ratio of consecutive terms:

2. limn→∞5n22n2+1\lim_{n \to \infty} \frac{5n^2}{2n^2 + 1}==

  1. 52\frac{5}{2}
  2. 00
  3. 55
  4. ∞\infty
Show the answer

Same degree: the ratio of the leading coefficients.

  • Correct: 52\frac{5}{2}:
  • 00:
  • 55:
  • ∞\infty:

Part 4 · See it

See it first

Two plots for n = 1 to 15, with blue bars for the terms and vermillion dots for the partial sums. Left: a sub n = n/(2n + 1); the bars level off near 1/2, not 0, and the partial sums climb by about 1/2 each step, past 7 by n = 15: the series diverges by the nth term test. Right: a sub n = 1/n; the bars shrink toward 0, yet the partial sums keep growing, past 3.3 by n = 15; terms approaching 0 do not settle the question, so the nth term test is silent.
Terms approaching 12\frac{1}{2} force the partial sums up without bound; terms approaching 0 leave the question open. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. If ∑an\sum a_n converges to SSSn→SS_n \to S and Sn−1→SS_{n-1} \to S, so ana_n=Sn−Sn−1→0= S_n - S_{n-1} \to 0
  2. So if liman\lim a_n is not 0, or does not existthe series cannot converge: it diverges
  3. If liman=0\lim a_n = 0the test says nothing; use another test

Part 6 · Key ideas

Key ideas

  • The nth term test: if limn→∞an≠0\lim_{n \to \infty} a_n \ne 0 (or the limit does not exist), then ∑an\sum a_n diverges.
  • It is a test for divergence only. When liman=0\lim a_n = 0 it gives no conclusion: ∑1n2\sum \frac{1}{n^2} converges and ∑1n\sum \frac{1}{n} diverges, and both have terms approaching 0.
  • Write it fully: the limit of the terms, its value, \"≠0\ne 0\", the test's name, and \"the series diverges\".

Part 7 · Misconception

A common mistake

The wrong idea: limn→∞1n=0\lim_{n \to \infty} \frac{1}{n} = 0, so ∑1n\sum \frac{1}{n} converges by the nth term test.

What actually happens: A limit of 0 means the nth term test says nothing. This series in fact diverges, as 10.5 shows.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

Terms of three series for large nn

Terms of the series ∑an\sum a_n, ∑bn\sum b_n and ∑cn\sum c_n, rounded to four places
nnan=2n2n2+5a_n = \frac{2n^2}{n^2 + 5}bn=1nb_n = \frac{1}{\sqrt{n}}cn=cos⁡(1n)c_n = \cos\left(\frac{1}{n}\right)
10101.90481.90480.31620.31620.99500.9950
1001001.99901.99900.10000.10001.00001.0000
100010002.00002.00000.03160.03161.00001.0000

1. Find limn→∞an\lim_{n \to \infty} a_n.

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

Divide the top and bottom by n2n^2: 21+5/n2→2\frac{2}{1 + 5/n^2} \to 2. The table agrees: 1.90481.9048, 1.99901.9990, 2.00002.0000.

  • Answer: 22

2. What does the nth term test say about ∑an\sum a_n?

  1. It converges to 2, since liman=2\lim a_n = 2
  2. Nothing, since the terms approach a limit
  3. It converges, since the terms level off
  4. It diverges, since liman=2≠0\lim a_n = 2 \ne 0
Show the answer

limn→∞an\lim_{n \to \infty} a_n=2≠0= 2 \ne 0, so ∑an\sum a_n diverges by the nth term test.

  • It converges to 2, since liman=2\lim a_n = 2: The limit of the terms is not the sum. Adding terms near 2 forever grows without bound.
  • Nothing, since the terms approach a limit: The test decides divergence exactly when that limit is not 0, as here.
  • It converges, since the terms level off: Terms leveling off at 2 add about 2 at each step, so the partial sums grow without bound.
  • Correct: It diverges, since liman=2≠0\lim a_n = 2 \ne 0: Right: terms that do not approach 0 make the series diverge.

3. What does the nth term test say about ∑bn\sum b_n?

  1. It converges, since limbn=0\lim b_n = 0
  2. It diverges, since bnb_n is positive
  3. Nothing: limbn=0\lim b_n = 0
  4. It diverges, since b1000b_{1000} is not 0
Show the answer

1n→0\frac{1}{\sqrt{n}} \to 0, so the nth term test gives no conclusion. (Topic 10.4 shows this series diverges.)

  • It converges, since limbn=0\lim b_n = 0: Terms approaching 0 are needed for convergence but do not guarantee it.
  • It diverges, since bnb_n is positive: Positive terms do not decide anything on their own.
  • Correct: Nothing: limbn=0\lim b_n = 0: Right: with terms approaching 0 the test is silent.
  • It diverges, since b1000b_{1000} is not 0: The test uses the limit of the terms, which here is 0, not any single term.

Graph

The terms of a series

The graph shows the terms an=3nn+4a_n = \frac{3n}{n + 4} of the series ∑n=1∞an\sum_{n=1}^{\infty} a_n, for n=1n = 1 to 1212. The dashed line is at height 3.
012345678910111200.511.522.533.5na_n

4. Which conclusion about ∑n=1∞an\sum_{n=1}^{\infty} a_n is correct?

  1. The series converges to 3
  2. The nth term test is inconclusive
  3. The series converges, since each ana_n is below 3
  4. The series diverges by the nth term test
Show the answer

limn→∞an\lim_{n \to \infty} a_n=3≠0= 3 \ne 0, so the series diverges by the nth term test.

  • The series converges to 3: 3 is the limit of the terms; a series whose terms approach 3 grows without bound.
  • The nth term test is inconclusive: It is inconclusive when the terms approach 0; here they approach 3.
  • The series converges, since each ana_n is below 3: Terms below 3 can still add without bound; these stay near 3.
  • Correct: The series diverges by the nth term test: Right: liman=3≠0\lim a_n = 3 \ne 0.

5. Which statement is the nth term test?

  1. If limn→∞an≠0\lim_{n \to \infty} a_n \ne 0, then ∑an\sum a_n diverges
  2. If limn→∞an=0\lim_{n \to \infty} a_n = 0, then ∑an\sum a_n converges
  3. If ∑an\sum a_n diverges, then limn→∞an≠0\lim_{n \to \infty} a_n \ne 0
  4. If limn→∞an=0\lim_{n \to \infty} a_n = 0, then ∑an\sum a_n diverges
Show the answer

If a series converges, its terms approach 0. Turned around: if the terms do not approach 0 (the limit is not 0 or does not exist), the series diverges.

  • Correct: If limn→∞an≠0\lim_{n \to \infty} a_n \ne 0, then ∑an\sum a_n diverges: Right: and if the limit is 0, the test says nothing.
  • If limn→∞an=0\lim_{n \to \infty} a_n = 0, then ∑an\sum a_n converges: This is false: terms approaching 0 are not enough (10.1 showed ∑ln⁡n+1n\sum \ln\frac{n + 1}{n} diverges).
  • If ∑an\sum a_n diverges, then limn→∞an≠0\lim_{n \to \infty} a_n \ne 0: This is the false converse: many divergent series have terms approaching 0.
  • If limn→∞an=0\lim_{n \to \infty} a_n = 0, then ∑an\sum a_n diverges: Terms approaching 0 leave the question open, in either direction.

6. What does the nth term test say about ∑n=1∞n2en\sum_{n=1}^{\infty} \frac{n^2}{e^n}?

  1. It diverges: the top and bottom both grow
  2. It converges: the terms approach 0
  3. Nothing: the terms approach 0
  4. It diverges: n2en\frac{n^2}{e^n} is positive
Show the answer

limn→∞n2en\lim_{n \to \infty} \frac{n^2}{e^n}=0= 0 (L'Hospital's rule twice), so the nth term test gives no conclusion.

  • It diverges: the top and bottom both grow: What matters is the limit of the quotient, which is 0.
  • It converges: the terms approach 0: A limit of 0 never proves convergence; another test is needed (a later test in this unit shows it converges).
  • Correct: Nothing: the terms approach 0: Right: ene^n outgrows n2n^2, so the test is silent.
  • It diverges: n2en\frac{n^2}{e^n} is positive: Positive terms decide nothing on their own.

Part 9 · Summary

Summary

If the terms of a series do not approach 0, the series diverges: that is the nth term test. If the terms do approach 0, the test is inconclusive and another test must decide. Say \"the series diverges\", give the limit of the terms, and show it is not 0.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic