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Infinite Series

Geometric Series

Definition of Geometric Series

Definition of Geometric Series

The series ∑n=0∞arn\sum_{n=0}^{\infty} ar^n is called a geometric series, where a≠0a \neq 0.

SymbolTypePronunciation/ExplanationMeaning in This Article
∑\sumGreek letterSigmaSummation symbol, representing series
∞\inftyMathematical symbolInfinityRepresents infinite series, infinite number of terms
rrMathematical symbolCommon ratioRatio between consecutive terms in geometric series

Convergence

Geometric Series Convergence

When ∣r∣<1|r| < 1, the series converges, and its sum is:

∑n=0∞arn=a1−r\sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r}

When ∣r∣≥1|r| \geq 1, the series diverges.

Proof
  1. Let Sn=a+ar+ar2+⋯+arn−1S_n = a + ar + ar^2 + \cdots + ar^{n-1}, then multiply by rr to get: rSn=ar+ar2+⋯+arnrS_n = ar + ar^2 + \cdots + ar^n
  2. Subtract the two equations: (1−r)Sn=a−arn=a(1−rn)(1 - r)S_n = a - ar^n = a(1 - r^n)
  3. If ∣r∣<1|r| < 1, then lim⁡n→∞rn=0\lim_{n \to \infty} r^n = 0, so: lim⁡n→∞Sn=a1−r\lim_{n \to \infty} S_n = \frac{a}{1 - r}
  4. When ∣r∣≥1|r| \geq 1, the partial sums do not converge, hence the series diverges.

Examples

Example 1

Determine the convergence of the series ∑n=0∞12n\sum_{n=0}^{\infty} \frac{1}{2^n} and find its sum if it converges.

Solution: This is a geometric series with a=1,r=12a = 1, r = \frac{1}{2}

Since ∣r∣=12<1|r| = \frac{1}{2} < 1, the series converges.

The sum is: S=a1−r=11−12=2S = \frac{a}{1 - r} = \frac{1}{1 - \frac{1}{2}} = 2

Example 2

Determine the convergence of the series ∑n=0∞13n\sum_{n=0}^{\infty} \frac{1}{3^n} and find its sum if it converges.

Solution: This is a geometric series with a=1,r=13a = 1, r = \frac{1}{3}

Since ∣r∣=13<1|r| = \frac{1}{3} < 1, the series converges.

The sum is: S=a1−r=11−13=32S = \frac{a}{1 - r} = \frac{1}{1 - \frac{1}{3}} = \frac{3}{2}

Example 3

Determine the convergence of the series ∑n=0∞2n\sum_{n=0}^{\infty} 2^n.

Solution: This is a geometric series with a=1,r=2a = 1, r = 2

Since ∣r∣=2≥1|r| = 2 \geq 1, the series diverges.

Exercises

Exercise 1

Determine the convergence of the series ∑n=0∞14n\sum_{n=0}^{\infty} \frac{1}{4^n} and find its sum if it converges.

geometric seriesseries convergenceseries summation

Approach: This is a geometric series; we need to determine the relationship between the absolute value of the common ratio and 1.

Steps:

  1. Identify the series type: ∑n=0∞14n\sum_{n=0}^{\infty} \frac{1}{4^n} is a geometric series
  2. Determine parameters: a=1,r=14a = 1, r = \frac{1}{4}
  3. Check convergence: ∣r∣=14<1|r| = \frac{1}{4} < 1, so the series converges
  4. Calculate the sum: S=a1−r=11−14=43S = \frac{a}{1-r} = \frac{1}{1-\frac{1}{4}} = \frac{4}{3}

Answer: The series converges with sum 43\frac{4}{3}.

Exercise 2

Determine the convergence of the series ∑n=0∞25n\sum_{n=0}^{\infty} \frac{2}{5^n} and find its sum if it converges.

geometric seriesseries convergenceseries summation

Approach: This is a geometric series; we need to determine the relationship between the absolute value of the common ratio and 1.

Steps:

  1. Identify the series type: ∑n=0∞25n\sum_{n=0}^{\infty} \frac{2}{5^n} is a geometric series
  2. Determine parameters: a=2,r=15a = 2, r = \frac{1}{5}
  3. Check convergence: ∣r∣=15<1|r| = \frac{1}{5} < 1, so the series converges
  4. Calculate the sum: S=a1−r=21−15=245=52S = \frac{a}{1-r} = \frac{2}{1-\frac{1}{5}} = \frac{2}{\frac{4}{5}} = \frac{5}{2}

Answer: The series converges with sum 52\frac{5}{2}.

Exercise 3

Determine the convergence of the series ∑n=0∞(−1)n\sum_{n=0}^{\infty} (-1)^n.

geometric seriesseries convergence

Approach: This is a geometric series; we need to determine the relationship between the absolute value of the common ratio and 1.

Steps:

  1. Identify the series type: ∑n=0∞(−1)n\sum_{n=0}^{\infty} (-1)^n is a geometric series
  2. Determine parameters: a=1,r=−1a = 1, r = -1
  3. Check convergence: ∣r∣=∣−1∣=1≥1|r| = |-1| = 1 \geq 1, so the series diverges

Answer: The series diverges.

Summary

Symbols Used in This Article

SymbolTypePronunciation/ExplanationMeaning in This Article
aaMathematical symbolFirst termFirst term of geometric series
SnS_nMathematical symbolPartial sumSum of first nn terms of the series
lim⁡\limMathematical symbolLimitRepresents limit of sequence or function
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