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

== Theorem ==
The Newton-Mercator series defines the natural logarithm function as a power series expansion:
valid for all $x \in \R$ such that $-1 < x \le 1$.
=== Corollary ===
== Proof ==
From Sum of Infinite Geometric Sequence, putting $-x$ for $x$:
:$(1): \quad \ds \sum_{n \mathop = 0}^\infty \paren {-x}^n = \frac 1 {1 + x}$
for $-1 < x < 1$.
From Power Series Converges Uniformly within Radius of Convergence, $(1)$ is uniformly convergent on every closed interval within the interval $\openint {-1} 1$.
From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:
== Sources ==
* : $\S 20$: Series for Exponential and Logarithmic Functions: $20.17$
* : $4.$ Mathematics: $4.2$ Series
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* : $\S 1.3.2$: Power series: $(1.41)$
* : $\S 1.2.9$: Generating Functions: $(24)$
* 
* 
* 
* 
* : $\S 22$: Taylor Series: Series for Exponential and Logarithmic Functions: $22.17.$
Category:Power Series Expansion for Logarithm of 1 + x
Category:Examples of Power Series
Category:Logarithms
Category:Newton-Mercator Series
Power Series Expansion for Logarithm of 1 + x/Corollary

== Corollary to Power Series Expansion for $\map \ln {1 + x}$ ==
valid for $-1 \le x < 1$.
== Proof ==
By Power Series Expansion for $\map \ln {1 + x}$:
:$\ds \map \ln {1 + x} = \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n$
Then:
Category:Power Series Expansion for Logarithm of 1 + x
Category:Examples of Power Series
Category:Logarithms
Power Series Expansion for Logarithm of 1 + x over 1 + x

== Theorem ==
where $H_n$ denotes the $n$th harmonic number:
:$H_n = \ds \sum_{r \mathop = 1}^n \dfrac 1 r = 1 + \dfrac 1 2 + \dfrac 1 3 \cdots + \dfrac 1 r$
valid for all $x \in \R$ such that $\size x < 1$.
== Proof ==
Let $\map f x = \dfrac {\map \ln {1 + x} } {1 + x}$.
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 $x = 0$.
From Nth Derivative of Natural Logarithm by Reciprocal:
:$\dfrac {\d^n} {\d x^n} \dfrac {\map \ln {1 + x} } {1 + x} = \paren {-1}^{n + 1} n! \dfrac {H_n - \map \ln {1 + x} } {\paren {1 + x}^{n + 1} }$
The result follows by setting $x = 0$ and substituting for $\map {f^{\paren n} } 0$ in $(1)$.
From the Ratio Test, we have:
If $\size x < 1 $, then $\ds \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} H_n x^n$ converges absolutely.
== Sources ==
* : $\S 20$: Miscellaneous Series: $20.51$
* : $\S 22$: Taylor Series: Miscellaneous Series: $22.51.$
Category:Examples of Power Series
Category:Logarithms
Power Series Expansion for Logarithm of Cosine of x

== Theorem ==
for all $x \in \R$ such that $\size x < \dfrac \pi 2$.
== Proof ==
From Power Series Expansion for Tangent Function:
for $\size x < \dfrac \pi 2$.
From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:
== Also presented as ==
== Sources ==
* : $\S 20$: Miscellaneous Series: $20.49$
* : $\S 22$: Taylor Series: Miscellaneous Series: $22.49.$
Category:Power Series Expansion for Logarithm of Cosine of x
Category:Examples of Power Series
Category:Logarithms
Category:Cosine Function
Power Series Expansion for Logarithm of Cosine of x/Also presented as

== Power Series Expansion for Logarithm of Cosine of x: Also presented as == 
The Power Series Expansion for Logarithm of Cosine of x can also be presented in the form:
where ${B_n}^*$ denotes the archaic form of the Bernoulli numbers.
== Sources ==
* : $\S 20$: Miscellaneous Series: $20.49$
* : $\S 22$: Taylor Series: Miscellaneous Series: $22.49.$
Category:Power Series Expansion for Logarithm of Cosine of x
Power Series Expansion for Logarithm of Gamma Function

== Theorem ==
The logarithm of the Gamma function has the power series expansion:
valid for all $x \in \C$ such that $\size x < 1$ and $x = 1$.
== Proof ==
When $x = 1$, we have $\sequence {a_k = \dfrac {\map \zeta k} k}_{k \mathop \ge 0}$ a decreasing sequence of positive terms in $\R$ which converges with a limit of zero.
By the Alternating Series Test, this alternating series converges at $x = 1$.
== Sources ==
* : Chapter $7$. Sums of Powers, Bernoulli Numbers, and the Gamma Function
Category:Euler-Mascheroni Constant
Category:Examples of Power Series
Category:Riemann Zeta Function
Power Series Expansion for Logarithm of Sine of x

== Theorem ==
for all $x \in \R$ such that $0 < \size x < \pi$.
== Proof ==
From Power Series Expansion for Cotangent Function:
for $0 < \size x < \pi$.
From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:
== Also presented as ==
== Sources ==
* : $\S 20$: Miscellaneous Series: $20.48$
* : $\S 22$: Taylor Series: Miscellaneous Series: $22.48.$
Category:Power Series Expansion for Logarithm of Sine of x
Category:Examples of Power Series
Category:Logarithms
Category:Sine Function
Power Series Expansion for Logarithm of Sine of x/Also presented as

== Power Series Expansion for Logarithm of Sine of x: Also presented as == 
The Power Series Expansion for Logarithm of Sine of x can also be presented in the form:
where ${B_n}^*$ denotes the archaic form of the Bernoulli numbers.
== Sources ==
* : $\S 20$: Miscellaneous Series: $20.48$
* : $\S 22$: Taylor Series: Miscellaneous Series: $22.48.$
Category:Power Series Expansion for Logarithm of Sine of x
Power Series Expansion for Logarithm of Tangent of x

== Theorem ==
for all $x \in \R$ such that $0 < \size x < \dfrac \pi 2$.
== Proof ==
From Power Series Expansion for Cosecant Function:
for $0 < \size {2 x} < \pi$.
From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:
for $0 < \size x < \dfrac \pi 2$.
== Also presented as ==
== Sources ==
* : $\S 20$: Miscellaneous Series: $20.50$
* : $\S 22$: Taylor Series: Miscellaneous Series: $22.50.$
Category:Power Series Expansion for Logarithm of Tangent of x
Category:Examples of Power Series
Category:Logarithms
Category:Tangent Function
Power Series Expansion for Logarithm of Tangent of x/Also presented as

== Power Series Expansion for Logarithm of Tangent of x: Also presented as == 
The Power Series Expansion for Logarithm of Tangent of x can also be presented in the form:
where ${B_n}^*$ denotes the archaic form of the Bernoulli numbers.
== Sources ==
* : $\S 20$: Miscellaneous Series: $20.50$
* : $\S 22$: Taylor Series: Miscellaneous Series: $22.50.$
Category:Power Series Expansion for Logarithm of Tangent of x
Power Series Expansion for Logarithm of x/Formulation 1

== Theorem ==
valid for all $x \in \R_{>0}$.
== Proof ==
From Power Series Expansion for $\dfrac 1 2 \map \ln {\dfrac {1 + x} {1 - x} }$:
:$(1): \quad \ds \frac 1 2 \map \ln {\frac {1 + x} {1 - x} } = \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {2 n + 1}$
for $-1 < x < 1$.
Let $z = \dfrac {1 + x} {1 - x}$.
Then:
Then we have:
:$\ds \lim_{x \mathop \to 1^-} \dfrac {1 + x} {1 - x} \to +\infty$
and:
:$\ds \lim_{x \mathop \to -1^+} \dfrac {1 + x} {1 - x} \to 0$
Thus when $x \in \openint {-1} 1$ we have that $z \in \openint 0 \to$.
Thus, substituting $z$ for $\dfrac {1 + x} {1 - x}$ in $(1)$ gives the result.
== Sources ==
* : $\S 20$: Series for Exponential and Logarithmic Functions: $20.19$
* : $\S 22$: Taylor Series: Series for Exponential and Logarithmic Functions: $22.19.$
Category:Power Series Expansion for Logarithm of x
Power Series Expansion for Logarithm of x/Formulation 2

== Theorem ==
valid for all $x \in \R$ such that $x \ge \dfrac 1 2$.
== Proof ==
From :
Let $z = \dfrac 1 {1 - x}$.
Then:
Then we have:
:$\ds \lim_{x \mathop \to 1^-} \dfrac 1 {1 - x} \to +\infty$
and:
:$\ds \lim_{x \mathop \to -1^+} \dfrac 1 {1 - x} \to \frac 1 2$
Thus when $x \in \hointr {-1} 1$ we have that $z \in \hointr {\dfrac 1 2} \to$.
Thus, substituting $z$ for $\dfrac 1 {1 - x}$ in $(1)$ gives the result.
== Sources ==
* : $\S 20$: Series for Exponential and Logarithmic Functions: $20.20$
* : $\S 22$: Taylor Series: Series for Exponential and Logarithmic Functions: $22.20.$
Category:Power Series Expansion for Logarithm of x
Power Series Expansion for Real Arccosecant Function

== Theorem ==
The arccosecant function has a Taylor Series expansion:
which converges for $\size x \ge 1$.
== Proof ==
From Arcsine of Reciprocal equals Arccosecant:
:$\arccsc x = \arcsin \dfrac 1 x$
From Power Series Expansion for Real Arcsine Function:
:$\ds \arcsin x = \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}$
which is converges for $\size x \le 1$.
The result follows by subtituting $\dfrac 1 x$ for $x$.
This converges for $\size {\dfrac 1 x} \le 1$, that is, for $\size x \ge 1$.
== Also see ==
* Power Series Expansion for Real Arcsine Function
* Power Series Expansion for Real Arccosine Function
* Power Series Expansion for Real Arctangent Function
* Power Series Expansion for Real Arccotangent Function
* Power Series Expansion for Real Arcsecant Function
== Sources ==
* : $\S 20$: Series for Trigonometric Functions: $20.32$
* : $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.32.$
Category:Examples of Power Series
Category:Arccosecant Function
Power Series Expansion for Real Arccosine Function

== Theorem ==
The arccosine function has a Taylor Series expansion:
which converges for $-1 \le x \le 1$.
== Proof ==
It follows from Power Series Expansion for Real Arcsine Function that the series is convergent for $-1 \le x \le 1$.
== Also see ==
* Power Series Expansion for Real Arcsine Function
* Power Series Expansion for Real Arctangent Function
* Power Series Expansion for Real Arccotangent Function
* Power Series Expansion for Real Arcsecant Function
* Power Series Expansion for Real Arccosecant Function
== Sources ==
* : $\S 20$: Series for Trigonometric Functions: $20.28$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* : $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.28.$
Category:Examples of Power Series
Category:Arccosine Function
Power Series Expansion for Real Arccotangent Function

== Theorem ==
The arccotangent function has a Taylor series expansion:
:$\arccot x = \begin {cases} \ds \frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1} & : -1 \le x \le 1 \\ \\
\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : x \ge 1 \\ \\
\ds \pi + \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : x \le -1
\end {cases}$
That is:
:$\arccot x = \begin {cases} \dfrac \pi 2 - \paren {x - \dfrac {x^3} 3 + \dfrac {x^5} 5 - \dfrac {x^7} 7 + \cdots} & : -1 \le x \le 1 \\ \\
\dfrac 1 x - \dfrac 1 {3 x^3} + \dfrac 1 {5 x^5} - \cdots & : x \ge 1 \\ \\
\pi + \dfrac 1 x - \dfrac 1 {3 x^3} + \dfrac 1 {5 x^5} - \cdots & : x \le -1
\end {cases}$
== Proof ==
From Sum of Arctangent and Arccotangent:
:$\arccot x = \dfrac \pi 2 - \arctan x$
The result follows from Power Series Expansion for Real Arctangent Function.
== Also see ==
* Power Series Expansion for Real Arcsine Function
* Power Series Expansion for Real Arccosine Function
* Power Series Expansion for Real Arctangent Function
* Power Series Expansion for Real Arcsecant Function
* Power Series Expansion for Real Arccosecant Function
== Sources ==
* : $\S 20$: Series for Trigonometric Functions: $20.30$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* : $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.30.$
Category:Examples of Power Series
Category:Arccotangent Function
Power Series Expansion for Real Arcsecant Function

== Theorem ==
The arcsecant function has a Taylor Series expansion:
which converges for $\size x \ge 1$.
== Proof ==
From Arccosine of Reciprocal equals Arcsecant:
:$\arcsec x = \arccos \dfrac 1 x$
From Power Series Expansion for Real Arccosine Function:
:$\ds \arccos x = \frac \pi 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}$
which is converges for $\size x \le 1$.
The result follows by subtituting $\dfrac 1 x$ for $x$.
This converges for $\size {\dfrac 1 x} \le 1$, that is, for $size x \ge 1$
== Also see ==
* Power Series Expansion for Real Arcsine Function
* Power Series Expansion for Real Arccosine Function
* Power Series Expansion for Real Arctangent Function
* Power Series Expansion for Real Arccotangent Function
* Power Series Expansion for Real Arccosecant Function
== Sources ==
* : $\S 20$: Series for Trigonometric Functions: $20.31$
* : $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.31.$
Category:Examples of Power Series
Category:Arcsecant Function
Power Series Expansion for Real Arcsine Function

== Theorem ==
The (real) arcsine function has a Taylor series expansion:
which converges for $-1 \le x \le 1$.
== Proof ==
From the General Binomial Theorem:
 
for $-1 < x < 1$.
From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:
We will now prove that the series converges for $-1 \le x \le 1$.
By Stirling's Formula:
Then:
Hence by Convergence of P-Series:
:$\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^{3/2} }$
is convergent.
So by the Comparison Test, the Taylor series is convergent for $-1 \le x \le 1$.
== Also see ==
* Power Series Expansion for Real Arccosine Function
* Power Series Expansion for Real Arctangent Function
* Power Series Expansion for Real Arccotangent Function
* Power Series Expansion for Real Arcsecant Function
* Power Series Expansion for Real Arccosecant Function
== Sources ==
* : $\S 20$: Series for Trigonometric Functions: $20.27$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* 
* : $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.27.$
Category:Examples of Power Series
Category:Arcsine Function
Power Series Expansion for Real Arctangent Function

== Theorem ==
The arctangent function has a Taylor series expansion:
:$\arctan x = \begin {cases} \ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1} & : -1 \le x \le 1 \\ \\
\ds \frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : x \ge 1 \\ \\
\ds -\frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : x \le -1
\end {cases}$
That is:
:$\arctan x = \begin {cases} x - \dfrac {x^3} 3 + \dfrac {x^5} 5 - \dfrac {x^7} 7 + \dfrac {x^9} 9 - \cdots & : -1 \le x \le 1 \\ \\
\dfrac \pi 2 - \dfrac 1 x + \dfrac 1 {3 x^3} - \dfrac 1 {5 x^5} + \cdots & : x \ge 1 \\ \\
-\dfrac \pi 2 - \dfrac 1 x + \dfrac 1 {3 x^3} - \dfrac 1 {5 x^5} + \cdots & : x \le -1
\end {cases}$
== Proof ==
From Sum of Infinite Geometric Sequence:
:$(1): \quad \ds \sum_{n \mathop = 0}^\infty \paren {-x^2}^n = \frac 1 {1 + x^2}$
for $-1 < x < 1$.
From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:
For $-1 \le x \le 1$, the sequence $\sequence {\dfrac {x^{2 n + 1}} {2 n + 1} }$ is decreasing and converges to zero.
Therefore the series converges in the given range by the Alternating Series Test.
Now consider the case $x \ge 1$:
We also have: 
Substituting $x$ for $-x$ gives us the expansion for $x \le -1$: 
== Also see ==
* Power Series Expansion for Real Arcsine Function
* Power Series Expansion for Real Arccosine Function
* Power Series Expansion for Real Arccotangent Function
* Power Series Expansion for Real Arcsecant Function
* Power Series Expansion for Real Arccosecant Function
== Sources ==
* : $\S 20$: Series for Trigonometric Functions: $20.29$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* : $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.29.$
Category:Examples of Power Series
Category:Arctangent Function
Power Series Expansion for Real Area Hyperbolic Cosecant

== Theorem ==
The (real) area hyperbolic cosecant function has a Taylor series expansion:
== Proof ==
From Power Series Expansion for Real Area Hyperbolic Sine:
From Real Area Hyperbolic Sine of Reciprocal equals Real Area Hyperbolic Cosecant:
:$\map \arsinh {\dfrac 1 x} = \arcsch x$
So:
Hence the result.
== Also see ==
* Power Series Expansion for Real Area Hyperbolic Sine
* Power Series Expansion for Real Area Hyperbolic Cosine
* Power Series Expansion for Real Area Hyperbolic Tangent
* Power Series Expansion for Real Area Hyperbolic Cotangent
* Power Series Expansion for Real Area Hyperbolic Secant
Category:Examples of Power Series
Category:Inverse Hyperbolic Cosecant
Power Series Expansion for Real Area Hyperbolic Cosine

== Theorem ==
The (real) area hyperbolic cosine function has a Taylor series expansion:
for $x \ge 1$.
== Proof ==
=== Lemma 1 ===
We have that the (real) area hyperbolic cosine is defined for $x \ge 1$.
Let $z = \dfrac 1 x$.
Then we have:
:$0 < \dfrac 1 z \le 1$
Now we consider:
== Also presented as ==
== Also see ==
* Power Series Expansion for Real Area Hyperbolic Sine
* Power Series Expansion for Real Area Hyperbolic Tangent
* Power Series Expansion for Real Area Hyperbolic Cotangent
* Power Series Expansion for Real Area Hyperbolic Secant
* Power Series Expansion for Real Area Hyperbolic Cosecant
== Sources ==
* : $\S 20$: Series for Hyperbolic Functions: $20.40$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* : $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.40.$
Category:Power Series Expansion for Real Area Hyperbolic Cosine
Category:Examples of Power Series
Category:Inverse Hyperbolic Cosine
Power Series Expansion for Real Area Hyperbolic Cosine/Also presented as

== Power Series Expansion for Real Area Hyperbolic Cosine: Also presented as ==
Some sources present this result in the form:
This takes into account the interpretation that $\cosh^{-1} x$ is a multifunction arising from the fact that $\cosh x = \map \cosh {-1}$ for $\size x \ge 1$.
== Sources ==
* : $\S 20$: Series for Hyperbolic Functions: $20.40$
* : $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.40.$
Category:Power Series Expansion for Real Area Hyperbolic Cosine
Power Series Expansion for Real Area Hyperbolic Cosine/Lemma 2

== Lemma for Power Series Expansion for Real Area Hyperbolic Cosine ==
:$\map \ln {1 + \sqrt {1 - x^2} } = \ln 2 - \paren {\dfrac 1 2 \cdot \dfrac {x^2} 2 + \dfrac {1 \times 3} {2 \times 4} \cdot \dfrac {x^4} 4 + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \cdot \dfrac {x^6} 6 + \cdots}$
This holds for $x \in \R: \size x < 1$.
== Proof ==
=== Lemma 1 ===
It remains to determine the value of the arbitrary constant $C$.
Let $x \to 0$.
Then we have:
Category:Power Series Expansion for Real Area Hyperbolic Cosine
Power Series Expansion for Real Area Hyperbolic Cosine/Mistake

== Source Work ==
:$8$. Taylor Series and Power Series:
::Appendix: Table $8.2$: Power Series of Important Functions
== Mistake ==
:$\cosh^{-1} x = \paren {\ln 2} x - \dfrac 1 {\paren {2 \times 2} x^2} - \dfrac {1 \times 3} {\paren {2 \times 4 \times 4} x^4} - \dfrac {1 \times 3 \times 5} {\paren {2 \times 4 \times 6 \times 6} x^6} - \cdots \quad \paren {x \ge 1}$
== Correction ==
The first term should read $\map \ln {2 x}$.
Hence the formula should be:
:$\cosh^{-1} x = \map \ln {2 x} - \dfrac 1 {\paren {2 \times 2} x^2} - \dfrac {1 \times 3} {\paren {2 \times 4 \times 4} x^4} - \dfrac {1 \times 3 \times 5} {\paren {2 \times 4 \times 6 \times 6} x^6} - \cdots$
== Sources ==
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
Category:Mistakes/Mathematics for Engineers and Scientists
Power Series Expansion for Real Area Hyperbolic Cotangent

== Theorem ==
The (real) area hyperbolic cotangent function has a Taylor series expansion:
for $\size x > 1$.
== Proof ==
From Power Series Expansion for Real Area Hyperbolic Tangent:
for $\size x < 1$.
From Real Area Hyperbolic Tangent of Reciprocal equals Real Area Hyperbolic Cotangent:
:$\map \artanh {\dfrac 1 x} = \arcoth x$
So:
Hence the result.
== Also see ==
* Power Series Expansion for Real Area Hyperbolic Sine
* Power Series Expansion for Real Area Hyperbolic Cosine
* Power Series Expansion for Real Area Hyperbolic Tangent
* Power Series Expansion for Real Area Hyperbolic Secant
* Power Series Expansion for Real Area Hyperbolic Cosecant
== Sources ==
* : $\S 20$: Series for Hyperbolic Functions: $20.42$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* : $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.42.$
Category:Examples of Power Series
Category:Inverse Hyperbolic Cotangent
Power Series Expansion for Real Area Hyperbolic Secant

== Theorem ==
The (real) area hyperbolic secant function has a Taylor series expansion:
for $0 < x \le 1$.
== Proof ==
From Power Series Expansion for Real Area Hyperbolic Cosine:
for $x \ge 1$.
From Real Area Hyperbolic Cosine of Reciprocal equals Real Area Hyperbolic Secant:
:$\map \arcosh {\dfrac 1 x} = \arsech x$
So:
For $1 \le \dfrac 1 x$ we have that $0 < x \le 1$.
Hence the result.
== Also see ==
* Power Series Expansion for Real Area Hyperbolic Sine
* Power Series Expansion for Real Area Hyperbolic Cosine
* Power Series Expansion for Real Area Hyperbolic Tangent
* Power Series Expansion for Real Area Hyperbolic Cotangent
* Power Series Expansion for Real Area Hyperbolic Cosecant
Category:Examples of Power Series
Category:Inverse Hyperbolic Secant
Power Series Expansion for Real Area Hyperbolic Sine

== Theorem ==
The (real) area hyperbolic sine function has a Taylor series expansion:
== Proof ==
=== Lemma 1 ===
From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:
We will now prove that the series converges for $-1 < x < 1$.
By Stirling's Formula:
Then:
Hence by Convergence of P-Series:
:$\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^{3/2} }$
is convergent.
So by the Comparison Test, the Taylor series is convergent for $-1 \le x \le 1$.
Another lemma:
=== Lemma $2$ ===
Let $x \ge 1$.
Let $z = \dfrac 1 x$.
Then we have:
:$0 < \dfrac 1 z \le 1$
Now we consider:
Now let $x \le -1$.
We have that Inverse Hyperbolic Sine is Odd Function.
That is:
:$\arsinh x = -\map \arsinh {-x}$
Thus for $x \le -1$:
Hence the result.
== Also presented as ==
This can also be presented as:
where $\pm$ is $+$ for $x \ge 1$ and $-$ for $x \le -1$.
== Also see ==
* Power Series Expansion for Real Area Hyperbolic Cosine
* Power Series Expansion for Real Area Hyperbolic Tangent
* Power Series Expansion for Real Area Hyperbolic Cotangent
* Power Series Expansion for Real Area Hyperbolic Secant
* Power Series Expansion for Real Area Hyperbolic Cosecant
== Sources ==
* : $\S 20$: Series for Hyperbolic Functions: $20.39$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* : $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.39.$
* 
for a citation that looks like:
Category:Power Series Expansion for Real Area Hyperbolic Sine
Category:Examples of Power Series
Category:Inverse Hyperbolic Sine
Power Series Expansion for Real Area Hyperbolic Sine/Lemma 2

== Lemma for Power Series Expansion for Real Area Hyperbolic Sine ==
:$\map \ln {1 + \sqrt {1 + x^2} } = \ln 2 + \dfrac 1 2 \cdot \dfrac {x^2} 2 - \dfrac {1 \times 3} {2 \times 4} \cdot \dfrac {x^4} 4 + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \cdot \dfrac {x^6} 6 - \cdots$
This holds for $x \in \R: \size x < 1$.
== Proof ==
It remains to determine the value of the arbitrary constant $C$.
Let $x \to 0$.
Then we have:
Category:Power Series Expansion for Real Area Hyperbolic Sine