avewright/proofwiki-math huggingface dataset:
Theorem: Exponent Combination Laws/Power of Product

:$\paren {a b}^x = a^x b^x$

Proof:
{{begin-eqn}}
{{eqn | l = \paren {a b}^x
      | r = \map \exp {x \map \ln {a b} }
      | c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {x \ln a + x \ln b}
      | c = Sum of Logarithms
}}
{{eqn | r = \map \exp {x \ln a} \map \exp {x \ln b}
      | c = Exponential of Sum
}}
{{eqn | r = a^x b^x
      | c = {{Defof|Power to Real Number}}
}}
{{end-eqn}}
{{qed}}
Theorem: Exponent Combination Laws/Negative Power

: $a^{-x} = \dfrac 1 {a^x}$

Proof:
{{begin-eqn}}
{{eqn | l = a^{-x}
      | r = \map \exp {-x \ln a}
      | c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \paren {\map \exp {x \ln a} }^{-1}
      | c = Exponential of Product
}}
{{eqn | r = \frac 1 {\map \exp {x \ln a} }
      | c = 
}}
{{eqn | r = \frac 1 {a^x}
      | c = {{Defof|Power to Real Number}}
}}
{{end-eqn}}
{{qed}}
Theorem: Exponent Combination Laws/Power of Power

Let $x, y \in \R$ be real numbers.
Let $a^x$ be defined as $a$ to the power of $x$.
Then:
:$\paren {a^x}^y = a^{x y}$

Proof:
{{begin-eqn}}
{{eqn | l = a^{x y}
      | r = \map \exp {x y \ln a}
      | c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {y \, \map \ln {a^x} }
      | c = Logarithms of Powers
}}
{{eqn | r = \paren {a^x}^y
      | c = {{Defof|Power to Real Number}}
}}
{{end-eqn}}
{{qed}}
Theorem: Exponent Combination Laws/Quotient of Powers

:$\dfrac{a^x} {a^y} = a^{x - y}$

Proof:
{{begin-eqn}}
{{eqn | l = \frac {a^x} {a^y}
      | r = a^x \paren {\frac 1 {a^y} }
      | c = 
}}
{{eqn | r = \paren {a^x} \paren {a^{-y} }
      | c = Exponent Combination Laws: Negative Power
}}
{{eqn | r = a^{x - y}
      | c = Product of Powers
}}
{{end-eqn}}
{{qed}}
Theorem: Exponent Combination Laws/Power of Quotient

:$\paren {\dfrac a b}^x = \dfrac {a^x} {b^x}$

Proof:
{{begin-eqn}}
{{eqn | l = \paren {\frac a b}^x
      | r = \map \exp {x \, \map \ln {\frac a b} }
      | c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {x \ln a - x \ln b}
      | c = Sum of Logarithms
}}
{{eqn | r = \frac {\map \exp {x \ln a} } {\map \exp {x \ln b} }
      | c = Exponential of Sum
}}
{{eqn | r = \frac {a^x} {b^x}
      | c = {{Defof|Power to Real Number}}
}}
{{end-eqn}}
{{qed}}
Category:Exponent Combination Laws
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Theorem: Exponent Combination Laws/Negative Power of Quotient

:$\paren {\dfrac a b}^{-x} = \paren {\dfrac b a}^x$

Proof:
{{begin-eqn}}
{{eqn | l = \paren {\frac a b} ^{-x}
      | r = \paren {\frac 1 {\paren {\frac a b} } }^x
      | c = Exponent Combination Laws: Negative Power
}}
{{eqn | r = \paren {\frac b a}^x
      | c = 
}}
{{end-eqn}}
{{qed}}
Category:Exponent Combination Laws
obg84cper7fdgxo5j5vlbmc7opho5sd
Theorem: Derivative of General Exponential Function

Let $a \in \R_{>0}$.
Let $a^x$ be $a$ to the power of $x$.
Then:
:$\map {\dfrac \d {\d x} } {a^x} = a^x \ln a$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {a^x}
      | r = \map {\frac \d {\d x} } {e^{x \ln a} }
      | c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \ln a e^{x \ln a}
      | c = Derivative of $e^{a x}$
}}
{{eqn | r = a^x \ln a
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of General Exponential Function

Let $a \in \R_{>0}$.
Let $a^x$ be $a$ to the power of $x$.
Then:
:$\map {\dfrac \d {\d x} } {a^x} = a^x \ln a$

Proof:
{{begin-eqn}}
{{eqn | l = \lim_{h \mathop \to 0} \frac {a^{x + h} - a^x} h
      | r = a^x \lim_{h \mathop \to 0} \frac {a^h - 1} h
      | c = Product of Powers
}}
{{eqn | r = a^x \lim_{h \mathop \to 0} \frac {\map \exp {h \ln a} - 1} h
      | c = {{Defof|Power to Real Number}}
}}
{{eqn | r = a^x \lim_{h \mathop \to 0} \paren {\frac {\map \exp {h \ln a} - 1} {h \ln a} } \paren {\frac {h \ln a} h}
      | c = 
}}
{{eqn | r = a^x \lim_{h \mathop \to 0} \paren {\frac {h \ln a} h}
      | c = Derivative of Exponential at Zero
}}
{{eqn | r = a^x \ln a
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of General Exponential Function

Let $a \in \R_{>0}$.
Let $a^x$ be $a$ to the power of $x$.
Then:
:$\map {\dfrac \d {\d x} } {a^x} = a^x \ln a$

Proof:
Let $y = a^x$.
Then:
{{begin-eqn}}
{{eqn | l = \ln y
      | r = x \ln a
      | c = Logarithm of Power
}}
{{eqn | ll= \leadsto
      | l = \dfrac 1 y \dfrac {\d y} {\d x}
      | r = \ln a
      | c = Derivative of Identity Function: Corollary
}}
{{eqn | ll= \leadsto
      | l = \dfrac 1 {a^x} \dfrac {\d y} {\d x}
      | r = \ln a
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \dfrac {\d y} {\d x}
      | r = a^x \ln a
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Binomial Theorem/General Binomial Theorem

Let $\alpha \in \R$ be a real number.
Let $x \in \R$ be a real number such that $\size x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {1 + x}^\alpha
      | r = \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n
      | c = 
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \dbinom \alpha n x^n
      | c = 
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {\prod_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } x^n
      | c = 
}}
{{eqn | r = 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } {2!} x^2 + \dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 2} } {3!} x^3 + \cdots
      | c = 
}}
{{end-eqn}}
where:
:$\alpha^{\underline n}$ denotes the falling factorial
:$\dbinom \alpha n$ denotes a binomial coefficient.

Proof:
Let $R$ be the radius of convergence of the power series:
:$\ds \map f x = \sum_{n \mathop = 0}^\infty \frac {\prod \limits_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } {n!} x^n$
Then:
{{begin-eqn}}
{{eqn | l = \frac 1 R
      | r = \lim_{n \mathop \to \infty} \frac {\size {\alpha \paren {\alpha - 1} \dotsm \paren {\alpha - n} } } {\paren {n + 1}!} \frac {n!} {\size {\alpha \paren {\alpha - 1} \dotsm \paren {\alpha - n + 1} } }
      | c = Radius of Convergence from Limit of Sequence
}}
{{eqn | r = \lim_{n \mathop \to \infty} \frac {\size {\alpha - n} } {n + 1}
      | c = 
}}
{{eqn | r = 1
      | c = 
}}
{{end-eqn}}
Thus for $\size x < 1$, Power Series is Differentiable on Interval of Convergence applies:
:$\ds D_x \map f x = \sum_{n \mathop = 1}^\infty \frac {\prod \limits_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } {n!} n x^{n - 1}$
This leads to:
{{begin-eqn}}
{{eqn | l = \paren {1 + x} D_x \map f x
      | r = \sum_{n \mathop = 1}^\infty \frac {\prod \limits_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } {\paren {n - 1}!} x^{n - 1} + \sum_{n \mathop = 1}^\infty \frac {\prod \limits_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } {\paren {n - 1}!} x^n
      | c = 
}}
{{eqn | r = \alpha + \sum_{n \mathop = 1}^\infty \paren {\frac {\prod \limits_{k \mathop = 0}^n \paren {\alpha - k} } {n!} + \frac {\prod \limits_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } {\paren {n - 1}!} } x^n
      | c = 
}}
{{eqn | r = \alpha + \sum_{n \mathop = 1}^\infty \frac {\prod \limits_{k \mathop = 0}^n \paren {\alpha - k} } {\paren {n - 1}!} \paren {\frac 1 n + \frac 1 {\alpha - n} } x^n
      | c = 
}}
{{eqn | r = \alpha + \sum_{n \mathop = 1}^\infty \frac {\prod \limits_{k \mathop = 0}^n \paren {\alpha - k} } {\paren {n - 1}!} \frac \alpha {n \paren {\alpha - n} } x^n
      | c = 
}}
{{eqn | r = \alpha \paren {1 + \sum_{n \mathop = 1}^\infty \frac {\prod \limits_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } {n!} x^n}
      | c = 
}}
{{eqn | r = \alpha \map f x
      | c = 
}}
{{end-eqn}}
Gathering up:
:$\paren {1 + x} D_x \map f x = \alpha \map f x$
Thus:
:$\map {D_x} {\dfrac {\map f x} {\paren {1 + x}^\alpha} } = -\alpha \paren {1 + x}^{-\alpha - 1} \map f x + \paren {1 + x}^{-\alpha} D_x \map f x = 0$
So $\map f x = c \paren {1 + x}^\alpha$ when $\size x < 1$ for some constant $c$.
But $\map f 0 = 1$ and hence $c = 1$.
{{qed}}
Theorem: Binomial Theorem/General Binomial Theorem

Let $\alpha \in \R$ be a real number.
Let $x \in \R$ be a real number such that $\size x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {1 + x}^\alpha
      | r = \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n
      | c = 
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \dbinom \alpha n x^n
      | c = 
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {\prod_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } x^n
      | c = 
}}
{{eqn | r = 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } {2!} x^2 + \dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 2} } {3!} x^3 + \cdots
      | c = 
}}
{{end-eqn}}
where:
:$\alpha^{\underline n}$ denotes the falling factorial
:$\dbinom \alpha n$ denotes a binomial coefficient.

Proof:
From Sum over k of r-kt choose k by r over r-kt by z^k:
:$\ds \sum_n \dbinom {\alpha - n t} k \dfrac \alpha {\alpha - n t} z^n = x^\alpha$
where:
:$z = x^{t + 1} - x^t$
:$x = 1$ for $z = 0$.
Setting $t = 0$:
{{begin-eqn}}
{{eqn | l = \sum_k \dbinom {\alpha - n \times 0} n \dfrac \alpha {\alpha - n \times 0} z^n
      | r = x^\alpha
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \sum_n \dbinom \alpha n \dfrac \alpha \alpha z^n
      | r = \paren {1 + z}^\alpha
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \sum_n \dbinom \alpha n z^n
      | r = \paren {1 + z}^\alpha
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Binomial Theorem/General Binomial Theorem

Let $\alpha \in \R$ be a real number.
Let $x \in \R$ be a real number such that $\size x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {1 + x}^\alpha
      | r = \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n
      | c = 
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \dbinom \alpha n x^n
      | c = 
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {\prod_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } x^n
      | c = 
}}
{{eqn | r = 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } {2!} x^2 + \dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 2} } {3!} x^3 + \cdots
      | c = 
}}
{{end-eqn}}
where:
:$\alpha^{\underline n}$ denotes the falling factorial
:$\dbinom \alpha n$ denotes a binomial coefficient.

Proof:
The series is the Maclaurin series expansion of the function $\map f x = \paren {1 + x}^\alpha$.
The derivatives of $f$ are:
{{begin-eqn}}
{{eqn | l = \map {f^{\paren 0} } x
      | r = \paren {1 + x}^\alpha
      | c = 
}}
{{eqn | l = \map {f^{\paren 1} } x
      | r = \alpha \paren {1 + x}^{\alpha - 1}
      | c = 
}}
{{eqn | l = \map {f^{\paren 2} } x
      | r = \alpha \paren {\alpha - 1} \paren {1 + x}^{\alpha - 2}
      | c = 
}}
{{eqn | l = \map {f^{\paren n} } x
      | r = \alpha \paren {\alpha - 1} \cdots \paren {\alpha - n + 1} \paren {1 + x}^{\alpha - n}
      | c = 
}}
{{eqn | r = \alpha^{\underline n} \paren {1 + x}^{\alpha - n}
      | c = 
}}
{{end-eqn}}
Evaluated at $x = 0$, we have:
{{begin-eqn}}
{{eqn | l = \map {f^{\paren 0} } x
      | r = \alpha^{\underline n} \paren {1 + 0}^{\alpha - n}
      | c = 
}}
{{eqn | r = \alpha^{\underline n}
      | c = 
}}
{{end-eqn}}
The Maclaurin series of $f$ is:
{{begin-eqn}}
{{eqn | l = \map f x
      | r = \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \map {f^{\paren n} } 0
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \alpha^{\underline n}
      | c = substituting derivative at $0$
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n
      | c = rearranging
}}
{{end-eqn}}
{{qed}}
Theorem: Real Sine Function is Continuous

The real sine function $\sin: \R \to \R$ is continuous on $\R$.

Proof:
Recall the definition of the sine function:
:$\ds \forall x \in \R: \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$
Thus $\sin x$ is expressible in the form of a power series.
From Sine Function is Absolutely Convergent, we have that the interval of convergence of $\sin x$ is the whole of $\R$. 
From Power Series is Differentiable on Interval of Convergence, it follows that $\sin x$ is continuous on the whole of $\R$.
{{qed}}
Category:Sine Function
h1y2opwmi2s1hjui09y3ongommlag6u
Theorem: Sine Function is Absolutely Convergent

The real sine function $\sin: \R \to \R$ is absolutely convergent.

Proof:
Recall the definition of the sine function:
:$\ds \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$
For:
:$\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$
to be absolutely convergent we want:
:$\ds \sum_{n \mathop = 0}^\infty \size {\paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} } = \sum_{n \mathop = 0}^\infty \frac {\size x^{2 n + 1} } {\paren {2 n + 1}!}$
to be convergent.
But:
:$\ds \sum_{n \mathop = 0}^\infty \frac {\size x^{2 n + 1} } {\paren {2 n + 1}!}$
is just the terms of:
:$\ds \sum_{n \mathop = 0}^\infty \frac {\size x^n}{n!}$
for odd $n$.
Thus:
:$\ds \sum_{n \mathop = 0}^\infty \frac {\size x^{2 n + 1} } {\paren {2 n + 1}!} < \sum_{n \mathop = 0}^\infty \frac {\size x^n} {n!}$
But:
:$\ds \sum_{n \mathop = 0}^\infty \frac {\size x^n} {n!} = \exp \size x$
from the Taylor Series Expansion for Exponential Function of $\size x$, which converges for all $x \in \R$.
The result follows from the Squeeze Theorem.
{{qed}}
Theorem: Sine Function is Absolutely Convergent

The real sine function $\sin: \R \to \R$ is absolutely convergent.

Proof:
The definition of the complex sine function is:
:$\ds \forall z \in \C: \sin z = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!}$
By definition of absolutely convergent complex series, we must show that the power series
:$\ds \sum_{n \mathop = 0}^\infty \size {\paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!} }$
is convergent.
We have
{{begin-eqn}}
{{eqn | l = \ds \sum_{n \mathop = 0}^\infty \size {\paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!} }
      | r = \sum_{n \mathop = 0}^\infty \frac {\size z^{2 n + 1} } {\paren {2 n + 1}!}
      | c = {{Defof|Complex Modulus}} {{explain|If we need this step explained, then it needs to be broken out into a series of steps that illustrates that the complex modulus is absolutely multiplicative, or whatever, and of course that $\size {\paren {-1}^n} {{=}} 1$. That is, if we are going to bother doing that here.}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \paren{ \frac {\size z^{2 n + 1} } {\paren {2 n + 1}!} + \frac {\size z^{2 n } } {\paren {2 n }!} }
      | c = Squeeze Theorem for Complex Sequences
      | o = \le
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {\size z^n} {n!}
      | c = changing indices }}
{{eqn | r = \exp \size z
      | c = Taylor Series Expansion for Exponential Function }}
{{end-eqn}}
The result follows from Squeeze Theorem for Complex Sequences.
{{qed}}
Theorem: Sine Function is Absolutely Convergent

The real sine function $\sin: \R \to \R$ is absolutely convergent.

Proof:
Radius of Convergence of Power Series Expansion for Sine Function shows that the radius of convergence of the complex sine function is infinite.
Then Existence of Radius of Convergence of Complex Power Series shows that the complex sine function is absolutely convergent.
{{qed}}
Theorem: Sine of Zero is Zero

:$\sin 0 = 0$

Proof:
Recall the definition of the sine function:
:$\ds \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$
Thus:
:$\ds \sin 0 = 0 - \frac {0^3} {3!} + \frac {0^5} {5!} - \cdots = 0$
{{qed}}
Theorem: Sine Function is Odd

:$\map \sin {-z} = -\sin z$
That is, the sine function is odd.

Proof:
{{begin-eqn}}
{{eqn | l = \map \sinh {-x}
      | r = \frac {e^{-x} - e^{-\paren {-x} } } 2
      | c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac {e^{-x} - e^x} 2
}}
{{eqn | r = -\frac {e^x - e^{-x} } 2
}}
{{eqn | r = -\sinh x
}}
{{end-eqn}}
{{qed}}
Theorem: Sine Function is Odd

:$\map \sin {-z} = -\sin z$
That is, the sine function is odd.

Proof:
{{begin-eqn}}
{{eqn	| l = \map \sinh {-x}
	| r = -i \, \map \sin {-i x}
	| c = Hyperbolic Sine in terms of Sine
}}
{{eqn	| r = i \, \map \sin {i x}
	| c = Sine Function is Odd
}}
{{eqn	| r = -\sinh x
	| c = Hyperbolic Sine in terms of Sine
}}
{{end-eqn}}
{{qed}}
Theorem: Sine Function is Odd

:$\map \sin {-z} = -\sin z$
That is, the sine function is odd.

Proof:
Recall the definition of the sine function:
{{begin-eqn}}
{{eqn | l = \sin z
      | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!}
}}
{{eqn | r = z - \frac {z^3} {3!} + \frac {z^5} {5!} - \cdots
}}
{{end-eqn}}
From Sign of Odd Power, we have that:
:$\forall n \in \N: -\paren {z^{2 n + 1} } = \paren {-z}^{2 n + 1}$
The result follows directly.
{{qed}}
Theorem: Sine Function is Odd

:$\map \sin {-z} = -\sin z$
That is, the sine function is odd.

Proof:
Using the Sine of Difference formula:
{{begin-eqn}}
{{eqn | l = \map \sin {-b}
      | r = \map \sin {0 - b}
}}
{{eqn | r = \sin 0 \cos b - \cos 0 \sin b
}}
{{eqn | r = -\sin b
}}
{{end-eqn}}
{{qed}}
Theorem: Cosine Function is Continuous

:$\cos x$ is continuous on $\R$.

Proof:
Recall the definition of the cosine function:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$
Thus $\cos x$ is expressible in the form of a power series.
From Cosine Function is Absolutely Convergent, we have that the interval of convergence of $\cos x$ is the whole of $\R$. 
From Power Series is Differentiable on Interval of Convergence, it follows that $\cos x$ is continuous on the whole of $\R$.
{{qed}}
Category:Cosine Function
j3ew02p868bb7byfgyzk6dxoei0bha9
Theorem: Cosine Function is Absolutely Convergent

:$\cos x$ is absolutely convergent for all $x \in \R$.

Proof:
Recall the definition of the cosine function:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$
For:
:$\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}$ 
to be absolutely convergent, we want:
:$\ds \sum_{n \mathop = 0}^\infty \size {\paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} } = \sum_{n \mathop = 0}^\infty \frac {\size x^{2 n} } {\paren {2 n}!}$ 
to be convergent.
But:
:$\ds \sum_{n \mathop = 0}^\infty \frac {\size x^{2 n} } {\paren {2 n}!}$ 
is just the terms of:
:$\ds \sum_{n \mathop = 0}^\infty \frac {\size x^n} {n!}$
for even $n$.
Thus:
:$\ds \sum_{n \mathop = 0}^\infty \frac {\size x^{2 n} } {\paren {2 n}!} < \sum_{n \mathop = 0}^\infty \frac {\size x^n}{n!}$
But:
:$\ds \sum_{n \mathop = 0}^\infty \frac {\size x^n}{n!} = \exp \size x$ 
from the Taylor Series Expansion for Exponential Function of $\size x$, which converges for all $x \in \R$.
Also, the sequence of partial sums:
:$\ds \sum_{n \mathop = 0}^k \frac {\size x^{2 n} } {\paren {2 n}!}$
is increasing.
The result follows from an application of the Monotone Convergence Theorem (Real Analysis).
{{qed}}
Theorem: Cosine Function is Absolutely Convergent

:$\cos x$ is absolutely convergent for all $x \in \R$.

Proof:
The definition of the complex cosine function is:
:$\ds \cos z = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}$
By definition of absolutely convergent complex series, we must show that the power series
:$\ds \sum_{n \mathop = 0}^\infty \size {\paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!} }$
is convergent.
We have
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \size {\paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!} }
      | r = \sum_{n \mathop = 0}^\infty \frac {\size z^{2 n} } {\paren {2 n}!}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \paren{ \frac {\size z^{2 n} } {\paren {2 n}!} + \frac {\size z^{2 n + 1} } {\paren {2 n + 1}!} }
      | c = Squeeze Theorem for Complex Sequences
      | o = \leq
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {\size z^n} {n!}
      | c = changing indices }}
{{eqn | r = \exp \size z
      | c = Taylor Series Expansion for Exponential Function }}
{{end-eqn}}
The result follows from Squeeze Theorem for Complex Sequences.
{{qed}}
Theorem: Cosine Function is Absolutely Convergent

:$\cos x$ is absolutely convergent for all $x \in \R$.

Proof:
Radius of Convergence of Power Series Expansion for Cosine Function shows that the radius of convergence of the complex cosine function is infinite.
Then Existence of Radius of Convergence of Complex Power Series shows that the complex cosine function is absolutely convergent.
{{qed}}
Theorem: Cosine of Zero is One

:$\cos 0 = 1$

Proof:
Recall the definition of the cosine function:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$
Thus:
:$\cos 0 = 1 - \dfrac {0^2} {2!} + \dfrac {0^4} {4!} - \cdots = 1$
{{qed}}
Theorem: Cosine of Integer Multiple of Pi

:$\forall n \in \Z: \cos n \pi = \paren {-1}^n$

Proof:
Recall the definition of the cosine function:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$
From Cosine of Zero is One, we have that:
:$\cos 0 = 1$
This takes care of the case $n = 0$.
From Sine and Cosine are Periodic on Reals:
: $\cos \paren {x + 2 \pi}= \cos x$
and thus:
:$\forall m \in \Z: \cos \paren {x + 2 m \pi} = \cos x$.
The above observations combine to establish that:
:$\cos 2 m \pi = 1$
Again from Sine and Cosine are Periodic on Reals:
:$\cos \paren {x + \pi} = -\cos x$
Combining this with the above reasoning, it follows that:
:$\forall m \in \Z: \cos \paren {2 m + 1} \pi = -1$
Note that $\forall n \in \Z$:
:If $n$ is even, it is of the form $n = 2m$ for some $m \in \Z$, and so $\cos n \pi = 1$
:If $n$ is odd, it is of the form $n = 2 m + 1$ for some $m \in \Z$, and so $\cos n \pi = -1$
This leads to the desired conclusion:
:$\cos n \pi = \paren {-1}^n$
{{qed}}
Theorem: Cosine Function is Even

:$\map \cos {-z} = \cos z$
That is, the cosine function is even.

Proof:
Recall the definition of the cosine function:
{{begin-eqn}}
{{eqn | l = \cos z
      | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}
}}
{{eqn | r = 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \cdots
}}
{{end-eqn}}
From Even Power is Non-Negative:
:$\forall n \in \N: z^{2 n} = \paren {-z}^{2 n}$
The result follows.
{{qed}}
Theorem: Cosine Function is Even

:$\map \cos {-z} = \cos z$
That is, the cosine function is even.

Proof:
{{begin-eqn}}
{{eqn | l = \map \cos {-z}
      | r = \frac {e^{i \paren {-z} } + e^{-i \paren {-z} } } 2
      | c = Euler's Cosine Identity
}}
{{eqn | r = \frac {e^{i z} + e^{-i z} } 2
      | c = simplifying
}}
{{eqn | r = \cos z
      | c = Euler's Cosine Identity
}}
{{end-eqn}}
{{qed}}
Theorem: Cosine Function is Even

:$\map \cos {-z} = \cos z$
That is, the cosine function is even.

Proof:
{{begin-eqn}}
{{eqn | l = \map \cos {-z}
      | r = \map \cos {0 - z}
}}
{{eqn | r = \cos 0 \cos z + \sin 0 \sin z
      | c = Cosine of Difference
}}
{{eqn | r = \cos z
}}
{{end-eqn}}
{{qed}}
Theorem: Cosine Function is Even

:$\map \cos {-z} = \cos z$
That is, the cosine function is even.

Proof:
{{begin-eqn}}
{{eqn | l = \map \cosh {-x}
      | r = \frac {e^{-x} + e^{-\paren {-x} } } 2
      | c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac {e^{-x} + e^x} 2
}}
{{eqn | r = \frac {e^x + e^{-x} } 2
}}
{{eqn | r = \cosh x
}}
{{end-eqn}}
{{qed}}
Theorem: Cosine Function is Even

:$\map \cos {-z} = \cos z$
That is, the cosine function is even.

Proof:
{{begin-eqn}}
{{eqn	| l = \map \cosh {-x}
	| r = \map \cos {-i x}
	| c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn	| r = \map \cos {i x}
	| c = Cosine Function is Even
}}
{{eqn	| r = \cosh x
	| c = Hyperbolic Cosine in terms of Cosine
}}
{{end-eqn}}
{{qed}}