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Wize AP Calculus (BC) Textbook > Sequences and Series

Test of Divergence

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Test of Divergence (nth Term Limit Test)

Consider the series ∑n=1∞an.\sum\limits^\infty_{n=1}a_n.
If lim⁡n→∞an≠0\lim\limits_{n\rightarrow\infty}a_n\neq0 (or does not exist), then ∑n=1∞an\sum\limits^\infty_{n=1}a_n diverges.

Identifying Clues

  • the terms ana_n get larger
  • the terms ana_n oscillate back and forth

Watch Out!
BE CAREFUL!
If lim⁡n→∞an=0\lim\limits_{n\rightarrow\infty}a_n = 0, it does NOT mean that ∑n=1∞an\sum\limits^\infty_{n=1}a_n converges!

Counter example:
The harmonic series ∑n=1∞1n{\displaystyle \sum_{n=1}^\infty}\frac{1}{n}:
The limit of the terms lim⁡n→∞1n=0{\displaystyle \lim_{n\to\infty}}\frac{1}{n}=0, but it is known that the harmonic series does NOT converge.

Practice Question

Determine if the series ∑n=1∞nsin⁡1n\sum\limits^\infty_{n=1}n\sin\frac{1}{n} converges or not.

Practice Quesetion

Determine whether the series ∑n=1∞n3+5n−12n3−7{\displaystyle \sum_{n=1}^\infty}\frac{n^3+5n-1}{2n^3-7} converges or diverges.

Practice Question

Does the series {1, −1, 1, 2, −2, 1, 2, 3, −3, 1, 2, 3, 4, −4, ...}\left\{1,\ -1,\ 1,\ 2,\ -2,\ 1,\ 2,\ 3,\ -3,\ 1,\ 2,\ 3,\ 4,\ -4,\ ...\right\} converge or diverge?

Practice Question

Determine whether the series ∑n=1∞(−2)n(n+1)5{\displaystyle \sum_{n=1}^\infty}\frac{(-2)^n}{(n+1)^5} converges or diverges.