Exponential Series

Definition of Exponential Series

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

符号说明

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$\sum$	Greek letter	Sigma	Summation symbol, representing series
$\infty$	Mathematical symbol	Infinity	Represents infinite series, infinite number of terms
$n!$	Mathematical symbol	Factorial	n factorial, $n! = n \times (n-1) \times \cdots \times 1$
$e$	Mathematical symbol	Natural constant	Base of natural logarithm, approximately 2.71828

Convergence

Exponential Series Convergence

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

$\sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x$

证明

Using the ratio test:

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

$\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{x}{n+1} = 0 < 1$

Therefore, for any real number $x$ , the series converges.

Examples

Example 1

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

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

Therefore, the sum is: $e^1 = e$

Example 2

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

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

Therefore, the sum is: $e^2$

Example 3

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

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

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

Exercises

Exercise 1

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

Reference Answer (2 个标签)

exponential series series summation

Solution Approach: This is the exponential series, need to determine the value of $x$ .

Detailed Steps:

Identify series type: $\sum_{n=0}^{\infty} \frac{3^n}{n!} = \sum_{n=0}^{\infty} \frac{x^n}{n!}$ , where $x = 3$
This is the exponential series with $x = 3$
Calculate sum: $S = e^3$

Answer: The sum is $e^3$ .

Exercise 2

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

Reference Answer (2 个标签)

exponential series series summation

Solution Approach: This is the exponential series, need to determine the value of $x$ .

Detailed Steps:

Identify series type: $\sum_{n=0}^{\infty} \frac{(-2)^n}{n!} = \sum_{n=0}^{\infty} \frac{x^n}{n!}$ , where $x = -2$
This is the exponential series with $x = -2$
Calculate sum: $S = e^{-2} = \frac{1}{e^2}$

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

Exercise 3

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

Reference Answer (2 个标签)

exponential series series summation

Solution Approach: This is the exponential series, need to determine the value of $x$ .

Detailed Steps:

Identify series type: $\sum_{n=0}^{\infty} \frac{(\ln 2)^n}{n!} = \sum_{n=0}^{\infty} \frac{x^n}{n!}$ , where $x = \ln 2$
This is the exponential series with $x = \ln 2$
Calculate sum: $S = e^{\ln 2} = 2$

Answer: The sum is $2$ .

Summary

Symbols Appearing in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$x$	Mathematical symbol	Variable	Variable in the exponential series
$\lim$	Mathematical symbol	Limit	Represents the limit of a sequence or function

Chinese-English Glossary

Chinese Term	English Term	Phonetic	Explanation
Exponential series	exponential series	/ˌekspəˈnenʃəl ˈsɪəriːz/	Series of the form $\sum_{n=0}^{\infty} \frac{x^n}{n!}$
Natural constant	natural constant	/ˈnætʃərəl ˈkɒnstənt/	Base of natural logarithm $e$ , approximately 2.71828
Factorial	factorial	/fækˈtɔːriəl/	$n! = n \times (n-1) \times \cdots \times 1$
Convergence	convergence	/kənˈvɜːdʒəns/	Series partial sums have a finite limit
Ratio test	ratio test	/ˈreɪʃiəʊ test/	Method to determine convergence using ratio of adjacent terms

PreviousHarmonic SeriesLearn the definition, divergence, and proof methods of the harmonic series. NextLogarithmic SeriesLearn the definition, convergence, and applications of logarithmic series.

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