nyuuzyou/wikis huggingface dataset:
Power Series Expansion for Cube Root of 1 + x

== Theorem ==
Let $x \in \R$ such that $-1 < x \le 1$.
Then:
:$\sqrt [3] {1 + x} = 1 + \dfrac 1 3 x - \dfrac 2 {3 \times 6} x^2 + \dfrac {2 \times 5} {3 \times 6 \times 9} x^3 - \cdots$
== Proof ==
== Sources ==
* : $\S 20$: Binomial Series: $20.14$
* : $\S 22$: Taylor Series: Binomial Series: $22.14.$
Category:Examples of Power Series
Power Series Expansion for Error Function

== Theorem ==
where:
:$\erf$ denotes the error function
:$x$ is a real number. 
== Proof ==
== Sources ==
* : $\S 35$: Error Function $\ds \map \erf x = \frac 2 {\sqrt \pi} \int_0^x e^{-u^2} \rd u$: $35.1$
* 
Category:Examples of Power Series
Category:Error Function
Power Series Expansion for Exponential Function

== Theorem ==
Let $\exp x$ be the exponential function.
Then:
== Proof ==
From Higher Derivatives of Exponential Function, we have:
:$\forall n \in \N: \map {f^{\paren n} } {\exp x} = \exp x$
Since $\exp 0 = 1$, the Taylor series expansion for $\exp x$ about $0$ is given by:
:$\ds \exp x = \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$
From Radius of Convergence of Power Series over Factorial, we know that this power series expansion converges for all $x \in \R$.
From Taylor's Theorem, we know that
:$\ds \exp x = 1 + \frac x {1!} + \frac {x^2} {2!} + \cdots + \frac {x^{n - 1} } {\paren {n - 1}!} + \frac {x^n} {n!} \map \exp \eta$
where $0 \le \eta \le x$.
Hence:
So the partial sums of the power series converge to $\exp x$.
The result follows.
== Historical Note ==
== Sources ==
* : $\S 20$: Series for Exponential and Logarithmic Functions: $20.15$
* : $\text {3-4}$ Generating Functions: Example $\text {3-7}$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* : $\S 15.5$
* : $\S 1.3.2$: Power series: $(1.45)$
* : $\S 1.2.9$: Generating Functions: $(22)$
* 
* 
* 
* 
* : $\S 22$: Taylor Series: Series for Exponential and Logarithmic Functions: $22.15.$
Category:Exponential Series
Category:Exponential Function
Category:Examples of Power Series
Category:Taylor Series
Power Series Expansion for Exponential Function/Historical Note

== Historical Note on Power Series Expansion for Exponential Function ==
The power series expansion for $e^x$ was first established by in $1665$.
== Sources ==
* : $2 \cdotp 718 \, 281 \, 828 \, 459 \, 045 \, 235 \, 360 \, 287 \, 471 \, 352 \, 662 \, 497 \, 757 \, 247 \, 093 \, 699 \ldots$
* : $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ \ldots$
Category:Historical Notes
Power Series Expansion for Exponential Integral Function/Formulation 1

== Theorem ==
Let $\Ei: \R_{>0} \to \R$ denote the exponential integral function:
:$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Then:
where $\gamma$ denotes the Euler-Mascheroni constant.
== Proof ==
== Sources ==
* : $\S 35$: Exponential Integral $\ds \map \Ei x = \int_x^\infty \frac {e^{-u} } u \rd u$: $35.8$
Category:Power Series Expansion for Exponential Integral Function
Power Series Expansion for Exponential Integral Function/Formulation 2

== Theorem ==
Let $\Ei: \R_{>0} \to \R$ denote the exponential integral function:
:$\map \Ei x = \ds \int_{t \mathop \to -\infty}^{t \mathop = x} \frac {e^t} t \rd t$
Then:
where $\gamma$ denotes the Euler-Mascheroni constant.
== Proof ==
== Sources ==
* : Introduction: $3$. Auxiliary Functions and Arguments
Category:Power Series Expansion for Exponential Integral Function
Power Series Expansion for Exponential of Cosine of x

== Theorem ==
for all $x \in \R$.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 20$: Miscellaneous Series: $20.44$
* : $\S 22$: Taylor Series: Miscellaneous Series: $22.44.$
Category:Power Series Expansion for Exponential of Cosine of x
Category:Examples of Power Series
Category:Cosine Function
Category:Exponential Function
Power Series Expansion for Exponential of Cosine of x/Proof 1

== Theorem ==
== Proof ==
Let $\map f x = e^{\cos x}$.
Then:
By definition of Taylor series:
:$\map f x \sim \ds \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$
and so expanding about $\xi = 0$:
Category:Power Series Expansion for Exponential of Cosine of x
Power Series Expansion for Exponential of Sine of x

== Theorem ==
:$e^{\sin x} = 1 + x + \dfrac {x^2} 2 - \dfrac {x^4} 8 - \dfrac {x^5} {15} + \cdots$
for all $x \in \R$.
== Proof ==
Let $\map f x = e^{\sin x}$.
Then:
By definition of Taylor series:
:$\ds \map f x \sim \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$
and so expanding about $\xi = 0$:
No pattern is immediately apparent.
== Sources ==
* : $\S 20$: Miscellaneous Series: $20.43$
* : $\S 22$: Taylor Series: Miscellaneous Series: $22.43.$
Category:Examples of Power Series
Category:Sine Function
Category:Exponential Function
Power Series Expansion for Exponential of Tangent of x

== Theorem ==
:$e^{\tan x} = 1 + x + \dfrac {x^2} 2 + \dfrac {x^3} 2 + \dfrac {3 x^4} 8 + \cdots$
for all $x \in \R$ such that $\size x < \dfrac \pi 2$.
== Proof ==
Let $\map f x = e^{\tan x}$.
Then:
By definition of Taylor series:
:$\ds \map f x \sim \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$
This is to be expanded about $\xi = 0$.
Note that $\tan 0 = 0$ and $\sec 0 = 1$.
Thus:
Hence:
== Sources ==
* : $\S 20$: Miscellaneous Series: $20.45$
* : $\S 22$: Taylor Series: Miscellaneous Series: $22.45.$
Category:Examples of Power Series
Category:Tangent Function
Category:Exponential Function
Power Series Expansion for Exponential of x by Cosine of x

== Theorem ==
for all $x \in \R$.
== Proof ==
Let $\map f x = e^x \cos x$.
By definition of Maclaurin series:
:$(1): \quad \map f x \sim \ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \map {f^{\paren n} } 0$
where $\map {f^{\paren n} } 0$ denotes the $n$th derivative of $f$ $x$ evaluated at $x = 0$.
It remains to be shown that:
:$\map {f^{\paren n} } 0 = 2^{n / 2} \map \cos {\dfrac {n \pi} 4}$
The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\map {f^{\paren n} } x = 2^{n / 2} e^x \map \cos {x + \dfrac {n \pi} 4}$
=== Basis for the Induction ===
$\map P 0$ is the case:
Thus $\map P 0$ is seen to hold.
This is the basis for the induction.
=== Induction Hypothesis ===
Now it needs to be shown that, if $\map P k$ is true, where $k \ge 0$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
:$\map {f^{\paren k} } x = 2^{k / 2} e^x \cos \left({x + \dfrac {k \pi} 4}\right)$
from which it is to be shown that:
:$\map {f^{\paren {k + 1} } } x = 2^{\paren {k + 1} / 2} e^x \map \cos {x + \dfrac {\paren {k + 1} \pi} 4}$
=== Induction Step ===
This is the induction step:
So $\map P k \implies \map P {k + 1}$ and by the Principle of Mathematical Induction:
:$\forall n \in \Z_{\ge 0}: \map {f^{\paren n} } x = 2^{n / 2} e^x \map \cos {x + \dfrac {n \pi} 4}$
The result follows by setting $x = 0$ and substituting for $\map {f^{\paren n} } 0$ in $(1)$.
== Sources ==
* : $\S 20$: Miscellaneous Series: $20.47$
* : $\S 22$: Taylor Series: Miscellaneous Series: $22.47.$
Category:Examples of Power Series
Category:Cosine Function
Category:Exponential Function
Power Series Expansion for Exponential of x by Sine of x

== Theorem ==
for all $x \in \R$.
== Proof ==
Let $\map f x = e^x \sin x$.
By definition of Maclaurin series:
:$(1): \quad \map f x \sim \ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \map {f^{\paren n} } 0$
where $\map {f^{\paren n} } 0$ denotes the $n$th derivative of $f$ $x$ evaluated at $x = 0$.
It remains to be shown that:
:$\map {f^{\paren n} } 0 = 2^{n / 2} \map \sin {\dfrac {n \pi} 4}$
The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\map {f^{\paren n} } x = 2^{n / 2} e^x \, \map \sin {x + \dfrac {n \pi} 4}$
=== Basis for the Induction ===
$\map P 0$ is the case:
Thus $\map P 0$ is seen to hold.
This is the basis for the induction.
=== Induction Hypothesis ===
Now it needs to be shown that, if $\map P k$ is true, where $k \ge 0$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
:$\map {f^{\paren k} } x = 2^{k / 2} e^x \, \map \sin {x + \dfrac {k \pi} 4}$
from which it is to be shown that:
:$\map {f^{\paren {k + 1} } } x = 2^{\paren {k + 1} / 2} e^x \, \map \sin {x + \dfrac {\paren {k + 1} \pi} 4}$
=== Induction Step ===
This is the induction step:
So $\map P k \implies \map P {k + 1}$ and by the Principle of Mathematical Induction:
:$\forall n \in \Z_{\ge 0}: \map {f^{\paren n} } x = 2^{n / 2} e^x \, \map \sin {x + \dfrac {n \pi} 4}$
The result follows by setting $x = 0$ and substituting for $\map {f^{\paren n} } 0$ in $(1)$.
== Sources ==
* : $\S 20$: Miscellaneous Series: $20.46$
* : $\S 22$: Taylor Series: Miscellaneous Series: $22.46.$
Category:Examples of Power Series
Category:Sine Function
Category:Exponential Function
Power Series Expansion for Fresnel Cosine Integral Function

== Theorem ==
:$\ds \map {\operatorname C} x = \sqrt {\frac 2 \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{4 n + 1} } {\paren {4 n + 1} \paren {2 n}!}$
where $\operatorname C$ denotes the Fresnel cosine integral function.
== Proof ==
== Sources ==
* : $\S 35$: Fresnel Cosine Integral $\ds \map {\operatorname C} x = \sqrt {\frac 2 \pi} \int_0^x \cos u^2 \rd u$: $35.21$
Category:Fresnel Cosine Integral Function
Category:Examples of Power Series
Power Series Expansion for Fresnel Sine Integral Function

== Theorem ==
:$\ds \map {\operatorname S} x = \sqrt {\frac 2 \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{4 n + 3} } {\paren {4 n + 3} \paren {2 n + 1}!}$
where $\operatorname S$ denotes the Fresnel sine integral function.
== Proof ==
== Sources ==
* : $\S 35$: Fresnel Sine Integral $\ds \map {\operatorname S} x = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u$: $35.18$
Category:Fresnel Sine Integral Function
Category:Examples of Power Series
Power Series Expansion for General Exponential Function

== Theorem ==
Let $a \in \R_{> 0}$ be a (strictly) positive real number.
Then:
Then:
== Proof ==
By definition of a power to a real number:
:$a^x = \map \exp {x \ln a}$
As $x \ln a$ is itself a real number, we can use Power Series Expansion for Exponential Function:
substituting $x \ln a$ for $x$.
== Sources ==
* : $\S 20$: Series for Exponential and Logarithmic Functions: $20.16$
* : $\S 22$: Taylor Series: Series for Exponential and Logarithmic Functions: $22.16.$
Category:Exponential Function
Category:Examples of Power Series
Power Series Expansion for Half Logarithm of 1 + x over 1 - x

== Theorem ==
valid for all $x \in \R$ such that $-1 < x < 1$.
== Proof ==
From Power Series Expansion for $\map \ln {1 + x}$:
:$(1): \quad \ds \map \ln {1 + x} = \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n$
for $-1 < x \le 1$.
From :
:$(2): \quad \ds \map \ln {1 - x} = - \sum_{n \mathop = 1}^\infty \frac {x^n} n$
for $-1 < x < 1$.
Then we have:
Hence the result.
== Sources ==
* : $\S 20$: Series for Exponential and Logarithmic Functions: $20.18$
* : $\S 22$: Taylor Series: Series for Exponential and Logarithmic Functions: $22.18.$
Category:Examples of Power Series
Category:Logarithms
Power Series Expansion for Harmonic Numbers

== Theorem ==
The harmonic numbers have the power series expansion:
where $\zeta$ denotes the Riemann zeta function.
This is valid for all $x \in \C$ such that $\size x < 1$.
== Proof ==
Our reliance upon the Sum of Infinite Geometric Sequence above where $m = 1$ implies $\size x < 1$, where $\size x$ denotes the complex modulus of $x$.
When $m = 1$ and $x = 1$, we have Grandi's Series, a well known divergent series.
== Sources ==
* : Chapter $8$. Analogues of the Gamma Function
Category:Harmonic Numbers
Category:Examples of Power Series
Category:Riemann Zeta Function
Category:Taylor Series
Power Series Expansion for Hyperbolic Cosecant Function

== Theorem ==
The hyperbolic cosecant function has a Taylor series expansion:
where $B_n$ denotes the Bernoulli numbers.
This converges for $0 < \size x < \pi$.
== Proof ==
=== Convergence ===
By Combination Theorem for Limits of Real Functions we can deduce the following.
This is less than $1$ if $\size x < \pi$.
Hence by the Ratio Test, the outer radius of convergence is $\pi$
The principal part of the Laurent series is finite so converges for $x \ne 0$.
Thus we have the annulus of convergence to be $0 < \size x < \pi$.
== Also presented as ==
== Also see ==
* Power Series Expansion for Hyperbolic Sine Function
* Power Series Expansion for Hyperbolic Cosine Function
* Power Series Expansion for Hyperbolic Tangent Function
* Power Series Expansion for Hyperbolic Cotangent Function
* Power Series Expansion for Hyperbolic Secant Function
== Sources ==
* : $\S 20$: Series for Hyperbolic Functions: $20.38$
* : $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.38.$
Category:Power Series Expansion for Hyperbolic Cosecant Function
Category:Examples of Power Series
Category:Bernoulli Numbers
Category:Hyperbolic Cosecant Function
Power Series Expansion for Hyperbolic Cosecant Function/Also presented as

== Power Series Expansion for Hyperbolic Cosecant Function: Also presented as == 
The Power Series Expansion for Hyperbolic Cosecant Function can also be presented in the form:
where ${B_n}^*$ denotes the archaic form of the Bernoulli numbers.
== Sources ==
* : $\S 20$: Series for Hyperbolic Functions: $20.38$
* : $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.38.$
Category:Power Series Expansion for Hyperbolic Cosecant Function
Power Series Expansion for Hyperbolic Cosine Function

== Theorem ==
The hyperbolic cosine function has the power series expansion:
valid for all $x \in \R$.
== Proof ==
From Derivative of Hyperbolic Cosine:
:$\dfrac \d {\d x} \cosh x = \sinh x$
From Derivative of Hyperbolic Sine:
:$\dfrac \d {\d x} \sinh x = \cosh x$
Hence:
:$\dfrac {\d^2} {\d x^2} \cosh x = \cosh x$
and so for all $m \in \N$:
where $k \in \Z$.
This leads to the Maclaurin series expansion:
From Series of Power over Factorial Converges, it follows that this series is convergent for all $x$.
== Also see ==
* Power Series Expansion for Hyperbolic Sine Function
* Power Series Expansion for Hyperbolic Tangent Function
* Power Series Expansion for Hyperbolic Cotangent Function
* Power Series Expansion for Hyperbolic Secant Function
* Power Series Expansion for Hyperbolic Cosecant Function
== Sources ==
* : $\S 20$: Series for Hyperbolic Functions: $20.34$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* 
* 
* : $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.34.$
Category:Hyperbolic Cosine Function
Category:Examples of Power Series
Category:Taylor Series
Power Series Expansion for Hyperbolic Cotangent Function

== Theorem ==
The hyperbolic cotangent function has a Taylor series expansion:
where $B_{2 n}$ denotes the Bernoulli numbers.
This converges for $0 < \size x < \pi$.
== Proof ==
By Combination Theorem for Limits of Real Functions we can deduce the following:
This is less than $1$ :
:$\size x < \pi$
Hence by the Ratio Test, the series converges for $\size x < \pi$.
== Also presented as ==
== Also see ==
* Power Series Expansion for Hyperbolic Sine Function
* Power Series Expansion for Hyperbolic Cosine Function
* Power Series Expansion for Hyperbolic Tangent Function
* Power Series Expansion for Hyperbolic Secant Function
* Power Series Expansion for Hyperbolic Cosecant Function
== Sources ==
* : $\S 20$: Series for Hyperbolic Functions: $20.36$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* : $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.36.$
Category:Power Series Expansion for Hyperbolic Cotangent Function
Category:Examples of Power Series
Category:Bernoulli Numbers
Category:Hyperbolic Cotangent Function
Power Series Expansion for Hyperbolic Cotangent Function/Also presented as

== Power Series Expansion for Hyperbolic Cotangent Function: Also presented as == 
The Power Series Expansion for Hyperbolic Cotangent Function can also be presented in the form:
where ${B_n}^*$ denotes the archaic form of the Bernoulli numbers.
== Sources ==
* : $\S 20$: Series for Hyperbolic Functions: $20.36$
* : $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.36.$
Category:Power Series Expansion for Hyperbolic Cotangent Function
Power Series Expansion for Hyperbolic Cotangent Function/Mistake

== Source Work ==
:Chapter $20$: Taylor Series
::Series for Hyperbolic Functions: $20.36$
:Chapter $22$: Taylor Series
::Series for Hyperbolic Functions: $22.36$
== Mistake ==
:$\coth x = \dfrac 1 x + \dfrac x 3 - \dfrac {x^3} {45} + \dfrac {2 x^5} {945} + \cdots \dfrac {\paren {-1}^n 2^{2 n} B_n x^{2 n - 1} } {\paren {2 n}!} + \cdots \quad 0 < \size x < \pi$
== Correction ==
Basically correct, but for good form:
:there is a $+$ sign where it should technically be a $-$ sign
:there should be a $+$ or $-$ sign either side of the $\cdots$.
Indeed, would present this:
:$\coth x = \dfrac 1 x + \dfrac x 3 - \dfrac {x^3} {45} + \dfrac {2 x^5} {945} - \cdots + \dfrac {\paren {-1}^n 2^{2 n} {B_n}^* x^{2 n - 1} } {\paren {2 n}!} + \cdots \quad 0 < \size x < \pi$
where ${B_n}^*$ denotes the archaic form of the Bernoulli numbers.
This is because the $x^7$ term is actually negative.
Also note that $B_n$ in the original page uses the archaic form of the Bernoulli numbers definition, while uses the more modern $B_{2 n}$ form.
== Sources ==
* : $\S 20$: Series for Hyperbolic Functions: $20.36$
* : $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.36.$
Category:Mistakes/Mathematical Handbook of Formulas and Tables
Power Series Expansion for Hyperbolic Secant Function

== Theorem ==
The hyperbolic secant function has a Taylor series expansion:
where $E_{2 n}$ denotes the Euler numbers.
This converges for $\size x < \dfrac \pi 2$.
== Proof ==
By definition of the Euler numbers:
:$\ds \sech x = \sum_{n \mathop = 0}^\infty \frac {E_n x^n} {n!}$
From Odd Euler Numbers Vanish:
:$E_{2 k + 1} = 0$
for $k \in \Z$.
Hence the result.
== Also presented as ==
== Also see ==
* Power Series Expansion for Hyperbolic Sine Function
* Power Series Expansion for Hyperbolic Cosine Function
* Power Series Expansion for Hyperbolic Tangent Function
* Power Series Expansion for Hyperbolic Cotangent Function
* Power Series Expansion for Hyperbolic Cosecant Function
== Sources ==
* : $\S 20$: Series for Hyperbolic Functions: $20.37$
* : $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.37.$
Category:Power Series Expansion for Hyperbolic Secant Function
Category:Examples of Power Series
Category:Euler Numbers
Category:Hyperbolic Secant Function
Power Series Expansion for Hyperbolic Secant Function/Also presented as

== Power Series Expansion for Hyperbolic Secant Function: Also presented as == 
The Power Series Expansion for Hyperbolic Secant Function can also be presented in the form:
where ${E_n}^*$ denotes the alternative form of the Euler numbers.
== Sources ==
* : $\S 20$: Series for Hyperbolic Functions: $20.37$
* : $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.37.$
Category:Power Series Expansion for Hyperbolic Secant Function
Power Series Expansion for Hyperbolic Sine Function

== Theorem ==
The hyperbolic sine function has the power series expansion:
valid for all $x \in \R$.
== Proof ==
From Derivative of Hyperbolic Sine:
:$\dfrac \d {\d x} \sinh x = \cosh x$
From Derivative of Hyperbolic Cosine:
:$\dfrac \d {\d x} \cosh x = \sinh x$
Hence:
and so for all $m \in \N$:
where $k \in \Z$.
This leads to the Maclaurin series expansion:
From Series of Power over Factorial Converges, it follows that this series is convergent for all $x$.
== Also see ==
* Power Series Expansion for Hyperbolic Cosine Function
* Power Series Expansion for Hyperbolic Tangent Function
* Power Series Expansion for Hyperbolic Cotangent Function
* Power Series Expansion for Hyperbolic Secant Function
* Power Series Expansion for Hyperbolic Cosecant Function
== Sources ==
* : $\S 20$: Series for Hyperbolic Functions: $20.33$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* 
* 
* : $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.33.$
Category:Hyperbolic Sine Function
Category:Examples of Power Series
Category:Taylor Series
Power Series Expansion for Hyperbolic Tangent Function

== Theorem ==
The hyperbolic tangent function has a Taylor series expansion:
where $B_{2 n}$ denotes the Bernoulli numbers.
This converges for $\size x < \dfrac \pi 2$.
== Proof ==
From Power Series Expansion for Hyperbolic Cotangent Function:
:$(1): \quad \coth x = \ds \sum_{n \mathop = 0}^\infty \frac {2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}$
Then:
By Combination Theorem for Limits of Real Functions we can deduce the following.
This is less than $1$ :
:$\size x < \dfrac \pi 2$
Hence by the Ratio Test, the series converges for $\size x < \dfrac \pi 2$.
== Also presented as ==
== Also see ==
* Power Series Expansion for Hyperbolic Sine Function
* Power Series Expansion for Hyperbolic Cosine Function
* Power Series Expansion for Hyperbolic Cotangent Function
* Power Series Expansion for Hyperbolic Secant Function
* Power Series Expansion for Hyperbolic Cosecant Function
== Sources ==
* : $\S 20$: Series for Hyperbolic Functions: $20.35$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* : $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.35.$
Category:Power Series Expansion for Hyperbolic Tangent Function
Category:Examples of Power Series
Category:Bernoulli Numbers
Category:Hyperbolic Tangent Function
Power Series Expansion for Hyperbolic Tangent Function/Also presented as

== Power Series Expansion for Hyperbolic Tangent Function: Also presented as == 
The Power Series Expansion for Hyperbolic Tangent Function can also be presented in the form:
where ${B_n}^*$ denotes the archaic form of the Bernoulli numbers.
== Sources ==
* : $\S 20$: Series for Hyperbolic Functions: $20.35$
* : $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.35.$
Category:Power Series Expansion for Hyperbolic Tangent Function
Power Series Expansion for Integer Power of Exponential Function minus 1

== Theorem ==
Let $e^z$ denote the exponential function.
Then:
where $\ds {k \brace n}$ denotes a Stirling number of the second kind.
== Proof ==
== Sources ==
* : $\S 1.2.9$: Generating Functions: $(23)$
Category:Exponential Function
Category:Examples of Power Series
Category:Power Series Expansion for Integer Power of Exponential Function minus 1
Power Series Expansion for Integer Power of Exponential Function minus 1/Proof

== Theorem ==
== Proof ==
The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\ds \paren {e^z - 1}^n = n! \sum_{k \mathop \in \Z} {k \brace n} \frac {z^k} {k!}$
$\map P 0$ is the case:
Thus $\map P 0$ is seen to hold.
=== Basis for the Induction ===
$\map P 1$ is the case:
Thus $\map P 1$ is seen to hold.
This is the basis for the induction.
=== Induction Hypothesis ===
Now it needs to be shown that, if $\map P r$ is true, where $r \ge 1$, then it logically follows that $\map P {r + 1}$ is true.
So this is the induction hypothesis:
:$\ds \paren {e^z - 1}^r = r! \sum_{k \mathop \in \Z} {k \brace r} \frac {z^k} {k!}$
from which it is to be shown that:
:$\ds \paren {e^z - 1}^{r + 1} = \paren {r + 1}! \sum_{k \mathop \in \Z} {k \brace r + 1} \frac {z^k} {k!}$
=== Induction Step ===
This is the induction step:
So $\map P r \implies \map P {r + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
:$\forall n \in \Z_{\ge 0}: \ds \paren {e^z - 1}^n = n! \sum_{k \mathop \in \Z} {k \brace n} \frac {z^k} {k!}$
== Sources ==
* : $\S 1.2.9$: Generating Functions: Exercise $5$
Category:Power Series Expansion for Integer Power of Exponential Function minus 1