CalcRef

Sequences & Series · Convergence Tests

Divergence Test

lim⁡n→∞an≠0⇒∑an diverges\lim_{n \to \infty} a_n \neq 0 \Rightarrow \sum a_n \text{ diverges}

If the terms do not approach zero, the series diverges. CAUTION: If the limit IS zero, the test is inconclusive (the series may still diverge).

  • 2 worked examples
  • 18 in Sequences & Series

Worked examples

Does Σ n/(2n+1) converge?
  1. lim(n→∞) n/(2n+1) = 1/2 ≠ 0

Answer: Diverges by the divergence test.

1 more worked examplePremium

Can the divergence test determine if Σ 1/n converges?

Step-by-step solution locked.

Unlock the full working. CalcRef Premium opens the step-by-step solution above, and 141 worked solutions across the whole library - the intermediate lines, not just the answer. The formula statement, its conditions, its variables and the first worked example stay free on every page.
See PremiumBrowse the free reference

See all 18 Sequences & Series formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the Divergence Test?

If the terms do not approach zero, the series diverges. CAUTION: If the limit IS zero, the test is inconclusive (the series may still diverge). It is one of the sequences & series formulas in the CalcRef reference.

How do you use the Divergence Test?

Worked example. Does Σ n/(2n+1) converge? lim(n→∞) n/(2n+1) = 1/2 ≠ 0. Answer: Diverges by the divergence test.

Practice

Quiz yourself on Sequences & Series

Divergence Test is one of 18 Sequences & Series formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.

Take today's free quizAll 18 Sequences & Series formulas