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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:

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

几何解释

推论

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.

符号说明

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 seriesseries 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 seriesseries 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 seriesseries 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

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NextLogarithmic SeriesLearn the definition, convergence, and applications of logarithmic series.