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Infinite Series

Exponential Series

Definition of Exponential Series

Definition of Exponential Series

The series ∑n=0∞xnn!\sum_{n=0}^{\infty} \frac{x^n}{n!} is called the exponential series.

SymbolTypePronunciation/ExplanationMeaning in This Article
∑\sumGreek letterSigmaSummation symbol, representing series
∞\inftyMathematical symbolInfinityRepresents infinite series, infinite number of terms
n!n!Mathematical symbolFactorialn factorial, n!=n×(n−1)×⋯×1n! = n \times (n-1) \times \cdots \times 1
eeMathematical symbolNatural constantBase of natural logarithm, approximately 2.71828

Convergence

Exponential Series Convergence

For any real number xx, the series converges, and its sum is:

∑n=0∞xnn!=ex\sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x

Proof

Using the ratio test:

an+1an=xn+1(n+1)!⋅n!xn=xn+1\frac{a_{n+1}}{a_n} = \frac{x^{n+1}}{(n+1)!} \cdot \frac{n!}{x^n} = \frac{x}{n+1}

lim⁡n→∞an+1an=lim⁡n→∞xn+1=0<1\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{x}{n+1} = 0 < 1

Therefore, for any real number xx, the series converges.

Examples

Example 1

Find the sum of the series ∑n=0∞1n!\sum_{n=0}^{\infty} \frac{1}{n!}.

Solution: This is the exponential series with x=1x = 1.

Therefore, the sum is: e1=ee^1 = e

Example 2

Find the sum of the series ∑n=0∞2nn!\sum_{n=0}^{\infty} \frac{2^n}{n!}.

Solution: This is the exponential series with x=2x = 2.

Therefore, the sum is: e2e^2

Example 3

Find the sum of the series ∑n=0∞(−1)nn!\sum_{n=0}^{\infty} \frac{(-1)^n}{n!}.

Solution: This is the exponential series with x=−1x = -1.

Therefore, the sum is: e−1=1ee^{-1} = \frac{1}{e}

Exercises

Exercise 1

Find the sum of the series ∑n=0∞3nn!\sum_{n=0}^{\infty} \frac{3^n}{n!}.

exponential seriesseries summation

Approach: This is the exponential series, need to determine the value of xx.

Steps:

  1. Identify series type: ∑n=0∞3nn!=∑n=0∞xnn!\sum_{n=0}^{\infty} \frac{3^n}{n!} = \sum_{n=0}^{\infty} \frac{x^n}{n!}, where x=3x = 3
  2. This is the exponential series with x=3x = 3
  3. Calculate sum: S=e3S = e^3

Answer: The sum is e3e^3.

Exercise 2

Find the sum of the series ∑n=0∞(−2)nn!\sum_{n=0}^{\infty} \frac{(-2)^n}{n!}.

exponential seriesseries summation

Approach: This is the exponential series, need to determine the value of xx.

Steps:

  1. Identify series type: ∑n=0∞(−2)nn!=∑n=0∞xnn!\sum_{n=0}^{\infty} \frac{(-2)^n}{n!} = \sum_{n=0}^{\infty} \frac{x^n}{n!}, where x=−2x = -2
  2. This is the exponential series with x=−2x = -2
  3. Calculate sum: S=e−2=1e2S = e^{-2} = \frac{1}{e^2}

Answer: The sum is 1e2\frac{1}{e^2}.

Exercise 3

Find the sum of the series ∑n=0∞(ln⁡2)nn!\sum_{n=0}^{\infty} \frac{(\ln 2)^n}{n!}.

exponential seriesseries summation

Approach: This is the exponential series, need to determine the value of xx.

Steps:

  1. Identify series type: ∑n=0∞(ln⁡2)nn!=∑n=0∞xnn!\sum_{n=0}^{\infty} \frac{(\ln 2)^n}{n!} = \sum_{n=0}^{\infty} \frac{x^n}{n!}, where x=ln⁡2x = \ln 2
  2. This is the exponential series with x=ln⁡2x = \ln 2
  3. Calculate sum: S=eln⁡2=2S = e^{\ln 2} = 2

Answer: The sum is 22.

Summary

Symbols Used in This Article

SymbolTypePronunciation/ExplanationMeaning in This Article
xxMathematical symbolVariableVariable in the exponential series
lim⁡\limMathematical symbolLimitRepresents the limit of a sequence or function
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Logarithmic Series

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