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Step 16: Calculus 2

Sequences and convergence tests

Step 16 · Calculus 2 topics

Find limits of sequences, add up infinite series through their partial sums, and use the main tests to decide whether a series converges.

Sequences and their limits

A sequence is an endless list a₁, a₂, a₃, … It converges to L if its terms get, and stay, as close to L as you like. Limits of sequences follow the same algebra as limits of functions.

Example: A fraction

Find the limit of aₙ = (3n + 1) ÷ (n + 2).

  1. Divide the top and bottom by n: (3 + 1/n) ÷ (1 + 2/n).
  2. As n grows, 1/n and 2/n go to 0.

Answer3

Example: Geometric sequences

Does (0.9)ⁿ converge? Does (1.1)ⁿ?

  1. Multiplying by 0.9 again and again shrinks toward 0.
  2. Multiplying by 1.1 again and again grows without bound.

Answer(0.9)ⁿ converges to 0, and (1.1)ⁿ diverges

Series and partial sums

A series adds up the terms of a sequence. Its partial sum SNS_N adds the first N terms. The series converges if the partial sums converge, and its sum is their limit.

∑n=1∞an=lim⁡N→∞SN\sum_{n=1}^{\infty} a_n = \lim_{N \to \infty} S_N
Partial sums of 1/2 + 1/4 + 1/8 + … creep up toward 1.

Example: A geometric series

Find ∑n=0∞5(12)n\sum_{n=0}^{\infty} 5\left(\frac{1}{2}\right)^{n}.

  1. A geometric series with first term a and ratio r sums to a1−r\frac{a}{1 - r} when |r| < 1.
  2. Here a = 5 and r = 0.5: 5 ÷ 0.5 = 10.

Answer10

Example: Telescoping

Find ∑n=1∞(1n−1n+1)\sum_{n=1}^{\infty} \left(\frac{1}{n} - \frac{1}{n + 1}\right).

  1. The partial sums collapse: (1 − ½) + (½ − ⅓) + … leaves 1 − 1 ÷ (N + 1).
  2. That approaches 1.

Answer1

Convergence tests

TestUse it whenVerdict
DivergenceTerms do not go to 0Diverges
p-series∑1np\sum \frac{1}{n^{p}}Converges if p > 1
ComparisonTerms are like a known seriesSame as the known series
RatioFactorials or powers like 2ⁿConverges if the ratio's limit is under 1
AlternatingSigns alternate, sizes shrink to 0Converges

Example: The ratio test

Does ∑n2n\sum \frac{n}{2^{n}} converge?

  1. an+1an=n+12n\frac{a_{n+1}}{a_{n}} = \frac{n + 1}{2n}.
  2. That approaches one half, which is less than 1.

AnswerYes, it converges

Example: Comparison

Does ∑1n2+n\sum \frac{1}{n^{2} + n} converge?

  1. Each term is less than 1n2\frac{1}{n^{2}}.
  2. The p-series with p = 2 converges.

AnswerYes, it converges

Example: Alternating

Does ∑(−1)n+1n\sum \frac{(-1)^{n+1}}{n} converge?

  1. The signs alternate, and 1 ÷ n shrinks to 0.
  2. So the alternating series test applies, even though the harmonic series without the signs diverges.

AnswerYes, to ln 2

Try it yourself

Work each one on paper first, then open the answer to check.

  1. Find the limit of (5n² − 1) ÷ (2n² + n).

    Show answer

    52\frac{5}{2}

  2. Find ∑n=0∞(13)n\sum_{n=0}^{\infty} \left(\frac{1}{3}\right)^{n}.

    Show answer

    32\frac{3}{2}

  3. Does ∑nn+1\sum \frac{n}{n + 1} converge?

    Show answer

    No: its terms approach 1, not 0

  4. Does ∑1n3\sum \frac{1}{n^{3}} converge?

    Show answer

    Yes: it is a p-series with p = 3

  5. Does ∑1n\sum \frac{1}{\sqrt{n}} converge?

    Show answer

    No: it is a p-series with p = 0.5

  6. Does ∑3nn!\sum \frac{3^{n}}{n!} converge?

    Show answer

    Yes, by the ratio test: the ratio 3 ÷ (n + 1) goes to 0

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