exponential series
Definition
Noun: * A mathematical series that represents the expansion of the exponential function ( e^x ). It is the sum of terms of the form ( \frac{x^n}{n!} ) for ( n ) starting from 0 and going to infinity.
Usage
The term "exponential series" is used specifically in mathematics to refer to the infinite series representation of the exponential function. * The exponential series for ( e^x ) is ( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots ). * A key property of the exponential series is that its derivative is equal to itself.
Advanced Usage
- Convergence: The exponential series converges absolutely for all real (and complex) numbers ( x ).
- The radius of convergence for the exponential series is infinite.
- Complex Numbers: The series is fundamental in defining the exponential function for complex arguments, leading to Euler's formula.
- By substituting ( ix ) into the exponential series, one can derive the series for cosine and sine.
Variants and Related Words
- Exponential Function (n): The function ( e^x ), of which the exponential series is the representation.
- The exponential function is one of the most important functions in mathematics.
- Taylor Series (n): A more general series expansion of which the exponential series is a specific example (the Taylor series of ( e^x ) about zero).
- The exponential series is the Maclaurin (or Taylor) series for the exponential function.
- Power Series (n): A series of the form ( \sum a
- The exponential series is a power series with coefficients ( an = \frac{1}{n!} ).
Synonyms
- Series expansion of ( e^x ): A descriptive synonym.
- The series expansion of ( e^x ) is used frequently in calculus.
Related Idioms or Phrases
- (To) grow exponentially: This common idiom, derived from the properties of the exponential function, means to increase at an extremely rapid rate. It is conceptually related but does not directly reference the series itself.
- The virus spread exponentially in the first month.
Noun
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a series derived from the expansion of an exponential expression
