nyuuzyou/wikis huggingface dataset:
Power Law Delta Sequence

== Theorem ==
Let $\sequence {\map {\delta_n} x}$ be a sequence such that:
:$\ds \map {\delta_n} x := \frac {\size x^{\frac 1 n - 1}} {2 n}$
Then $\sequence {\map {\delta_n} x}_{n \mathop \in {\N_{>0} } }$ is a delta sequence.
That is, in the distributional sense it holds that:
:$\ds \lim_{n \mathop \to \infty} \map {\delta_n} x = \map \delta x$
or
:$\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} x \map \phi x \rd x = \map \delta \phi$
where $\phi \in \map \DD \R$ is a test function, $\delta$ is the Dirac delta distribution, and $\map \delta x$ is the abuse of notation, usually interpreted as an infinitely thin and tall spike with its area equal to $1$.
== Proof ==
Let $a \in \R_{> 0}$.
Then:
Furthermore:
:$\ds \forall a \in \R_{> 0} : \lim_{n \mathop \to \infty} a^{\frac 1 n} = 1$
Suppose $a, b \in \R_{> 0} : 0 < a < b$.
Then:
Analogously, suppose $a < b < 0$.
Then:
Let $\epsilon, a \in \R_{> 0}$ such that $\epsilon < a$.
Then:
$\epsilon$ is an arbitrary positive real number.
Hence, for every $\epsilon \in \R_{> 0}$ contributions from expressions with $\map \phi {\xi_+}$ and $\map \phi {\xi_-}$ vanish.
Suppose $\xi_\epsilon \ne 0$.
By Real Numbers are Densely Ordered:
:$\forall \epsilon \in \R_{> 0} : \exists \epsilon' \in \R_{> 0} : 0 < \epsilon' < \epsilon$
Then $\epsilon'$ we have that $\xi_\epsilon = \xi_{+'}$ or $\xi_\epsilon = \xi_{-'}$, where $\xi_{+'} \in \hointr {\epsilon'} a$ and $\xi_{-'} \in \hointl {-a} {-\epsilon'}$.
But from the result above, for every $\epsilon' \in \R_{> 0}$ contributions from expressions with $\map \phi {\xi_{+'}}$ and $\map \phi {\xi_{-'}}$ vanish.
Therefore, the only nonvanishing contribution can come from $\xi_\epsilon = 0$.
Taking the limit $a \to \infty$ in the above expressions yields the desired result.
Category:Examples of Delta Sequences
Category:Dirac Delta Distribution
Power Reduction Formulas

== Theorem ==
=== Square of Sine ===
=== Square of Cosine ===
=== Square of Tangent ===
=== Cube of Sine ===
=== Cube of Cosine ===
=== Fourth Power of Sine ===
=== Fourth Power of Cosine ===
=== Fifth Power of Sine ===
=== Fifth Power of Cosine ===
=== Sixth Power of Cosine ===
</onlyinclude>
=== Seventh Power of Cosine ===
</onlyinclude>
where $\sin, \cos$ and $\tan$ denote sine, cosine and tangent respectively.
=== Square of Hyperbolic Sine ===
=== Square of Hyperbolic Cosine ===
=== Cube of Hyperbolic Sine ===
=== Cube of Hyperbolic Cosine ===
=== Fourth Power of Hyperbolic Sine ===
=== Fourth Power of Hyperbolic Cosine ===
where $\sinh$ and $\cosh$ denote hyperbolic sine and hyperbolic cosine respectively.
== Comment ==
The identities for $\sin^m x$ and $\cos^n x$ can be useful for integrating expressions of the form:
:$\ds \int \sin^m x \ \cos^n x \rd x$
Category:Trigonometric Identities
Category:Hyperbolic Functions
Power Reduction Formulas/Cosine Squared

== Theorem ==
:$\cos^2 x = \dfrac {1 + \cos 2 x} 2$
where $\cos$ denotes cosine.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.54$
* : Appendix $12$: Trigonometric formulae: Double-angle formulae
* : Appendix $14$: Trigonometric formulae: Double-angle formulae
Category:Cosine Function
Power Reduction Formulas/Cosine to 4th

== Theorem ==
:$\cos^4 x = \dfrac {3 + 4 \cos 2 x + \cos 4 x} 8$
where $\cos$ denotes cosine.
== Proof 1 ==
== Proof 2 ==
== Also defined as ==
This result can often be seen as:
:$\cos^4 x = \dfrac 3 8 + \dfrac {\cos 2 x} 2 + \dfrac {\cos 4 x} 8$
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.58$
Category:Power Reduction Formula for 4th Power of Cosine
Category:Cosine Function
Power Reduction Formulas/Cosine to 5th

== Theorem ==
:$\cos^5 x = \dfrac {10 \cos x + 5 \cos 3 x + \cos 5 x} {16}$
where $\cos$ denotes cosine.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.60$
Category:Cosine Function
Power Reduction Formulas/Hyperbolic Cosine Cubed

== Theorem ==
:$\cosh^3 x = \dfrac {\cosh 3 x + 3 \cosh x} 4$
where $\cosh$ denotes hyperbolic cosine.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Also see ==
* Square of Hyperbolic Sine
* Square of Hyperbolic Cosine
* Cube of Hyperbolic Sine
* Fourth Power of Hyperbolic Sine
* Fourth Power of Hyperbolic Cosine
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.39$: Powers of Hyperbolic Functions
Category:Hyperbolic Cosine Function
Category:Cube of Hyperbolic Cosine
Power Reduction Formulas/Hyperbolic Cosine Squared

== Theorem ==
:$\cosh^2 x = \dfrac {\cosh 2 x + 1} 2$
where $\cosh$ denotes hyperbolic cosine.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Also see ==
* Square of Hyperbolic Sine
* Cube of Hyperbolic Sine
* Cube of Hyperbolic Cosine
* Fourth Power of Hyperbolic Sine
* Fourth Power of Hyperbolic Cosine
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.37$: Powers of Hyperbolic Functions
Category:Hyperbolic Cosine Function
Category:Square of Hyperbolic Cosine
Power Reduction Formulas/Hyperbolic Cosine to 4th

== Theorem ==
:$\cosh^4 x = \dfrac {3 + 4 \cosh 2 x + \cosh 4 x} 8$
where $\cos$ denotes hyperbolic cosine.
== Proof 1 ==
== Proof 2 == 
== Proof 3 ==
== Also see ==
* Square of Hyperbolic Sine
* Square of Hyperbolic Cosine
* Cube of Hyperbolic Sine
* Cube of Hyperbolic Cosine
* Fourth Power of Hyperbolic Sine
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.41$: Powers of Hyperbolic Functions
Category:Hyperbolic Cosine Function
Category:Fourth Power of Hyperbolic Cosine
Power Reduction Formulas/Hyperbolic Sine Cubed

== Theorem ==
:$\sinh^3 x = \dfrac {\sinh 3 x - 3 \sinh x} 4$
where $\sin$ denotes hyperbolic sine.
== Proof ==
== Also see ==
* Square of Hyperbolic Sine
* Square of Hyperbolic Cosine
* Cube of Hyperbolic Cosine
* Fourth Power of Hyperbolic Sine
* Fourth Power of Hyperbolic Cosine
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.38$: Powers of Hyperbolic Functions
Category:Hyperbolic Sine Function
Power Reduction Formulas/Hyperbolic Sine Squared

== Theorem ==
:$\sinh^2 x = \dfrac {\cosh 2 x - 1} 2$
where $\sinh$ and $\cosh$ denote hyperbolic sine and hyperbolic cosine respectively.
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Square of Hyperbolic Cosine
* Cube of Hyperbolic Sine
* Cube of Hyperbolic Cosine
* Fourth Power of Hyperbolic Sine
* Fourth Power of Hyperbolic Cosine
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.36$: Powers of Hyperbolic Functions
Category:Hyperbolic Sine Function
Category:Square of Hyperbolic Sine
Power Reduction Formulas/Hyperbolic Sine to 4th

== Theorem ==
:$\sinh^4 x = \dfrac {3 - 4 \cosh 2 x + \cosh 4 x} 8$
where $\sinh$ and $\cosh$ denote hyperbolic sine and hyperbolic cosine respectively.
== Proof ==
== Also see ==
* Square of Hyperbolic Sine
* Square of Hyperbolic Cosine
* Cube of Hyperbolic Sine
* Cube of Hyperbolic Cosine
* Fourth Power of Hyperbolic Cosine
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.40$: Powers of Hyperbolic Functions
Category:Hyperbolic Sine Function
Power Reduction Formulas/Sine Cubed

== Theorem ==
:$\sin^3 x = \dfrac {3 \sin x - \sin 3 x} 4$
where $\sin$ denotes sine.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.55$
Category:Sine Function
Category:Cube of Sine
Power Reduction Formulas/Sine Squared

== Theorem ==
:$\sin^2 x = \dfrac {1 - \cos 2 x} 2$
where $\sin$ and $\cos$ denote sine and cosine respectively.
== Proof ==
== Historical Note ==
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.53$
* : Appendix $12$: Trigonometric formulae: Double-angle formulae
* : Appendix $14$: Trigonometric formulae: Double-angle formulae
Category:Sine Function
Power Reduction Formulas/Sine to 4th

== Theorem ==
:$\sin^4 x = \dfrac {3 - 4 \cos 2 x + \cos 4 x} 8$
where $\sin$ and $\cos$ denote sine and cosine respectively.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.57$
Category:Sine Function
Power Reduction Formulas/Sine to 5th

== Theorem ==
:$\sin^5 x = \dfrac {10 \sin x - 5 \sin 3 x + \sin 5 x} {16}$
where $\sin$ denotes sine.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.59$
Category:Sine Function
Power Rule for Derivatives

== Theorem ==
Let $n \in \R$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined.
=== Corollary ===
== Proof ==
This can be done in sections.
=== Proof for Natural Number Index ===
=== Proof for Integer Index ===
=== Proof for Fractional Index ===
=== Proof for Rational Index ===
=== Proof for Real Number Index ===
== Historical Note ==
== Sources ==
* 
* 
* : Chapter $\text{VI}$: On the Seashore
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $1$.
* : $\text {II}$. Calculus: Differentiation
* : $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $2.$ Power function
* : Appendix $2$: Table of derivatives and integrals of common functions
* : Chapter $\text {A}.19$: Leibniz (1646-1716)
* 
* : Appendix: Table $1$: Derivatives
* 
* : Appendix: Table $1$: Derivatives
* : Chapter $8$: The System of the World: Leibniz
* : Appendix $6$: Derivatives
* : Appendix $7$: Derivatives
Category:Power Rule for Derivatives
Category:Derivatives
Power Rule for Derivatives/Corollary

== Corollary to Power Rule for Derivatives ==
Let $n \in \R$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {\dfrac \d {\d x} } {c x^n} = n c x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
== Proof ==
== Sources ==
* : $\S 13$: General Rules of Differentiation: $13.4$
Category:Power Rule for Derivatives
Power Rule for Derivatives/Fractional Index

== Theorem ==
Let $n \in \N_{>0}$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^{1 / n}$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined.
== Proof 1 ==
== Proof 2 ==
Category:Power Rule for Derivatives
Power Rule for Derivatives/Fractional Index/Proof 1

== Theorem ==
== Proof ==
Let $n \in \N_{>0}$.
Thus, let $\map f x = x^{1 / n}$.
From the definition of the power to a rational number, or alternatively from the definition of the root of a number, $\map f x$ is defined when $x \ge 0$.
(However, see the special case where $x = 0$.)
From Continuity of Root Function, $\map f x$ is continuous over the open interval $\openint 0 \infty$, but not at $x = 0$ where it is continuous only on the right.
Let $y > x$.
From Inequalities Concerning Roots:
:$\forall n \in \N_{>0}: X Y^{1 / n} \, \size {x - y} \le n X Y \, \size {x^{1 / n} - y^{1 / n} } \le Y X^{1 / n} \, \size {x - y}$
where $x, y \in \closedint X Y$.
Setting $X = x$ and $Y = y$, this reduces (after algebra) to:
:$\dfrac 1 {n y} y^{1 / n} \le \dfrac {y^{1 / n} - x^{1 / n} } {y - x} \le \dfrac 1 {n x} x^{1 / n}$
From the Squeeze Theorem for Functions, it follows that:
:$\ds \lim_{y \mathop \to x^+} \dfrac {y^{1 / n} - x^{1 / n} } {y - x} = \dfrac 1 {n x} x^{1 / n} = \dfrac 1 n x^{\dfrac 1 n - 1}$
A similar argument shows that the left hand limit is the same.
Thus the result holds for $\map f x = x^{1 / n}$.
== Sources ==
* : $\S 10.15 \ (1)$
Category:Power Rule for Derivatives
Power Rule for Derivatives/Historical Note

== Historical Note on Power Rule for Derivatives ==
The Power Rule for Derivatives was stated, without proof or explanation, by in his $1684$ article Nova Methodus pro Maximis et Minimis, published in .
 had established exactly the same result in a privately circulated paper of $1669$: , by investigating the nature of a function whose area under the graph is $x^m$.
== Sources ==
* : Chapter $\text {A}.19$: Leibniz ()
* : Chapter $8$: The System of the World: Leibniz
Category:Historical Notes
Power Rule for Derivatives/Integer Index

== Theorem ==
Let $n \in \Z$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined.
== Proof ==
When $n \ge 0$ we use the result for Natural Number Index.
Now let $n \in \Z: n < 0$.
Then let $m = -n$ and so $m > 0$.
Thus $x^n = \dfrac 1 {x^m}$.
== Sources ==
* : $\S 10.11 \ (2)$
Category:Power Rule for Derivatives
Power Rule for Derivatives/Natural Number Index

== Theorem ==
Let $n \in \N$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n-1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined.
== Proof by Binomial Theorem ==
== Proof by Induction ==
== Proof by Difference of Two Powers ==
Category:Power Rule for Derivatives
Power Rule for Derivatives/Natural Number Index/Proof by Binomial Theorem

== Theorem ==
== Proof ==
Let $\map f x = x^n$ for $x \in \R, n \in \N$.
By the definition of the derivative:
:$\ds \dfrac \d {\d x} \map f x = \lim_{h \mathop \to 0} \dfrac {\map f {x + h} - \map f x} h = \lim_{h \mathop \to 0} \dfrac {\paren {x + h}^n - x^n} h$
Using the binomial theorem this simplifies to:
== Sources ==
* : $\text {II}$. Calculus: Differentiation: Standard Differential Coefficients
Category:Power Rule for Derivatives
Power Rule for Derivatives/Natural Number Index/Proof by Difference of Two Powers

== Theorem ==
== Proof ==
Let $\map f x = x^n$ for $x \in \R, n \in \N$.
Let $a \in \R$.
By definition of the derivative:
:$\ds \map {f'} a = \lim_{x \mathop \to a} \frac {\map f x - \map f a} {x - a} = \lim_{x \mathop \to a} \frac {x^n - a^n} {x - a}$
=== Case $\text I$ ===
For $n = 0$ it is possible to do:
We have that:
:$0 \cdot x^{0 - 1} = 0$
for all $x \ne 0$.
So in this case:
:$\map {f'} x = n x^{n - 1}$
=== Case $\text {II}$ ===
For $n = 1$ we have:
:$\map f x = x$
From Derivative of Identity Function:
:$\map {f'} x = 1$
Then we note that:
:$1 \cdot x^{1 - 1} = 1$
So for the case $n = 1$:
:$\map {f'} x = n x^{n - 1}$
=== Case $\text {III}$ ===
$\R$ is a commutative ring, so for $n \ge 2$ it is possible to do:
This holds for all $a \in \R$, so:
:$\map {f'} x = n x^{n - 1}$
Category:Power Rule for Derivatives
Power Rule for Derivatives/Natural Number Index/Proof by Induction

== Theorem ==
== Proof ==
We will use the notation $D \map f x = \map {f'} x$ as it is convenient.
Let $n = 0$.
Then:
:$\forall x \in \R: x^n = 1$
Thus $\map f x$ is the constant function $\map {f_1} x$ on $\R$.
Thus from Derivative of Constant, $D \map f x = \map D {x^0} = 0 x^{-1}$, except where $x = 0$.
So the result holds for $n = 0$.
Let $n = 1$.
Then:
:$\forall x \in \R: \map f x = x^n = x$
Then from Derivative of Identity Function:
:$\map D x = 1 = 1 \cdot x^{1 - 1}$
So the result holds for $n = 1$.
Now assume $\map D {x^k} = k x^{k - 1}$.
Then by the Product Rule for Derivatives:
:$\map D {x^{k + 1} } = \map D {x^k x} = x^k \map D x + \map D {x^k} x = x^k \cdot 1 + k x^{k - 1} x = \paren {k + 1} x^k$
The result follows by induction.
== Sources ==
* : $\S 10.10$
Category:Power Rule for Derivatives
Power Rule for Derivatives/Rational Index

== Theorem ==
Let $n \in \Q$.
Let $f: \R \to \R$ be the real function defined as $f \left({x}\right) = x^n$.
Then:
:$\map {f'} x = n x^{n-1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined.
== Proof ==
Let $n \in \Q$, such that $n = \dfrac p q$ where $p, q \in \Z, q \ne 0$.
Then we have:
== Sources ==
* : $\S 10.15 \ (2)$
Category:Power Rule for Derivatives
Power Rule for Derivatives/Real Number Index

== Theorem ==
Let $n \in \R$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n-1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined.
== Proof 1 ==
== Proof 2 ==
Category:Power Rule for Derivatives
Power Rule for Derivatives/Real Number Index/Proof 1

== Theorem ==
== Proof ==
We are going to prove that $\map {f'} x = n x^{n - 1}$ holds for all real $n$.
To do this, we compute the limit $\ds \lim_{h \mathop \to 0} \frac {\paren {x + h}^n - x^n} h$:
Now we use the following results:
:$\ds \lim_{x \mathop \to 0} \frac {\exp x - 1} x = 1$ from Derivative of Exponential at Zero
:$\ds \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x = 1$ from Derivative of Logarithm at One
to obtain:
:$x^n \cdot \dfrac {e^{n \map \ln {1 + \frac h x} } - 1} {n \map \ln {1 + \dfrac h x} } \cdot \dfrac {n \map \ln {1 + \dfrac h x}} {\dfrac h x} \cdot \dfrac 1 x \to n x^{n - 1}$ as $h \to 0$
Hence the result.
Category:Power Rule for Derivatives
Power Rule for Derivatives/Real Number Index/Proof 2

== Theorem ==
== Proof ==
Note this proof does not hold for $x = 0$.
Let $y$ = $\map f x$.
Then $y = x^n$.
Then:
Using:
* Derivative of Composite Function
* Derivative of Constant Multiple
* Corollary to Primitive of Reciprocal
and taking the derivative of both sides with respect to $x$ gives:
:$\dfrac 1 y \dfrac {\d y} {\d x} = n \dfrac 1 x$
Multiplying both sides of the equation by $y$ yields:
:$\dfrac {\d y} {\d x} = n \dfrac y x$
Substituting $x^n$ for $y$:
:$\dfrac {\d y} {\d x} = n \dfrac {x^n} x$
From Quotient of Powers:
:$\dfrac {\d y} {\d x} = n x^{n - 1}$
Category:Power Rule for Derivatives
Power Series Converges Uniformly within Radius of Convergence

== Theorem ==
Let $\ds S := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$.
Let $R$ be the radius of convergence of $S$.
Let $\rho \in \R$ such that $0 \le \rho < R$.
Then $S$ is uniformly convergent on $D = \set {x: \size {x - \xi} \le \rho}$.
== Proof ==
We shall make use of the Weierstrass M-Test to prove this result.
To begin with, for each $n \in N$, define for $x \in D$:
:$\map {f_n} x = a_n \paren {x - \xi}^n$
We have:
Define $M_n = \size {a_n \rho^n}$.
Then $M_n$ is an upper bound for $\size {\map {f_n} x}$.
So, by the definition of a supremum:
:$\ds \sup_{x \mathop \in D} \size {\map {f_n} x} \le M_n$
Next, to apply the Weierstrass M-Test, it remains to be shown that $\ds \sum_{n \mathop = 0}^\infty M_n = \sum_{n \mathop = 0}^\infty \size {a_n \rho^n}$ converges.
Choose $x = \xi + \rho$ and note that:
:$\size {x - \xi} = \size \rho = \rho < R$
Hence, by definition of radius of convergence, $x$ falls within the interval of convergence of $S$.
Thus $\ds S = \sum_{n \mathop = 0}^\infty a_n (x - \xi)^n = \sum_{n \mathop = 0}^\infty a_n \rho^n$ converges.
By Existence of Interval of Convergence of Power Series, a power series always converges absolutely at all points in its interval of convergence.
Hence $\ds \sum_{n \mathop = 0}^\infty M_n = \sum_{n \mathop = 0}^\infty \size {a_n \rho^n}$ also converges.
Finally then, having satisfied all the requirements to use Weierstrass M-Test, we can conclude that $\ds \sum_{n \mathop = 0}^\infty \map {f_n} x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ converges uniformly in $D$.
== Sources ==
* : $\S 1.3.2$: Power series
Category:Power Series
Power Series Converges to Continuous Function

== Theorem ==
Let $\ds \map f x := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$.
Let $R$ be the radius of convergence of $S$.
Then $\map f x$ is a continuous function on $\set {x: \size {x - \xi} < R}$.
== Proof ==
Let $\rho \in \R$ such that $0 \le \rho < R$.
From Power Series Converges Uniformly within Radius of Convergence, $\map f x$ is uniformly convergent on $\set {x: \size {x - \xi} \le \rho}$.
From Real Polynomial Function is Continuous, each of $\map {f_n} x = a_n x^n$ is a continuous function of $x$.
The result follows from Uniformly Convergent Series of Continuous Functions is Continuous.
== Sources ==
* : $\S 1.3.2$: Power series
Category:Power Series
Power Series Expansion for Complementary Error Function

== Theorem ==
:$\ds \map \erfc x = 1 - \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {n! \paren {2 n + 1} }$
where:
:$\erfc$ is the complementary error function
:$x$ is a real number.
== Proof ==
== Sources ==
* : $\S 35$: Complementary Error Function $\ds\map \erfc x = 1 - \map \erf x = \frac 2 {\sqrt \pi} \int_x^\infty e^{-u^2} \rd u$: $35.4$
Category:Examples of Power Series
Category:Complementary Error Function
Power Series Expansion for Cosecant Function

== Theorem ==
The cosecant function has a Laurent 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 ==
== Sources ==
* : $\S 20$: Series for Trigonometric Functions: $20.26$
* : $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.26.$
Category:Power Series Expansion for Cosecant Function
Category:Examples of Power Series
Category:Bernoulli Numbers
Category:Cosecant Function
Power Series Expansion for Cosecant Function/Also presented as

== Power Series Expansion for Cosecant Function: Also presented as == 
The Power Series Expansion for 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 Trigonometric Functions: $20.26$
* : $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.26.$
Category:Power Series Expansion for Cosecant Function
Power Series Expansion for Cosine Function

== Theorem ==
The cosine function has the power series expansion:
valid for all $x \in \R$.
== Proof ==
From Derivative of Cosine Function:
:$\dfrac \d {\d x} \cos x = -\sin x$
From Derivative of Sine Function:
:$\dfrac \d {\d x} \sin x = \cos x$
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 Sine Function
== Sources ==
* : $\S 20$: Series for Trigonometric Functions: $20.22$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* : $\S 1.3.2$: Power series: $(1.47)$
* 
* 
* : $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.22.$
Category:Cosine Function
Category:Examples of Power Series
Category:Taylor Series
Power Series Expansion for Cosine Integral Function plus Logarithm

== Theorem ==
:$\ds \map \Ci x = -\gamma - \ln x + \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} \frac {x^{2 n} } {\paren {2 n} \times \paren {2 n}!}$
where: 
:$\Ci$ denotes the cosine integral function
:$\gamma$ denotes the Euler-Mascheroni constant
:$x$ is a strictly positive real number.
== Proof ==
== Sources ==
* : $\S 35$: Cosine Integral $\ds \map \Ci x = \int_x^\infty \frac {\cos u} u \rd u$: $35.15$
Category:Cosine Integral Function
Category:Examples of Power Series
Power Series Expansion for Cotangent Function

== Theorem ==
The (real) 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 ==
* Laurent Series Expansion for Cotangent Function
== Sources ==
* : $\S 20$: Series for Trigonometric Functions: $20.24$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* : $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.24.$
Category:Power Series Expansion for Cotangent Function
Category:Examples of Power Series
Category:Bernoulli Numbers
Category:Cotangent Function
Power Series Expansion for Cotangent Function/Also presented as

== Power Series Expansion for Cotangent Function: Also presented as == 
The Power Series Expansion for 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 Trigonometric Functions: $20.24$
* : $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.24.$
Category:Power Series Expansion for Cotangent Function