Power Series

Examples

\begin{matrix} \sum_{κ = 0}^{\infty} x^{κ} & \sum_{κ = 0}^{\infty} {(-1)}^{κ} \frac{x^{κ}}{κ} & \sum_{κ = 0}^{\infty} {(\frac{1}{2})}^{κ} {(x - 2)}^{κ} \\ \sum_{κ = 0}^{\infty} \frac{x^{κ}}{κ!} & \sum_{κ = 0}^{\infty} κ! x^{κ} & \sum_{κ = 0}^{\infty} \frac{{(x - 2)}^{κ}}{(κ + 1) 3^{κ}} \end{matrix}

Terminology

Power series

A power series about $𝒶$ is a series of the form

\sum_{κ = 0}^{\infty} 𝒸_{κ} {(x - 𝒶)}^{κ} = 𝒸_{0} + 𝒸_{1} (x - 𝒶) + 𝒸_{2} {(x - 𝒶)}^{2} + \dots

The value $𝒶$ is the center and $𝒸_{κ}$ are the coefficients of the power series.

Convergence

The convergence of $\sum_{κ = 0}^{\infty} 𝒸_{κ} {(x - 𝒶)}^{κ}$ satisfies one of the:

$\exists 𝓇 \in ℝ^{+}$ such that
series converges for $| x - 𝒶 | < 𝓇$

series diverges for $| x - 𝒶 | > 𝓇$

may or may not converge at $x = 𝒶 - 𝓇$ and $x = 𝒶 + 𝓇$
converges absolutely for any $x$
converges only at $x = 𝒶$

Operations

Linearity

Let $\sum_{κ = 0}^{\infty} 𝒸_{κ} x^{κ}$ and $\sum_{κ = 0}^{\infty} 𝒹_{κ} x^{κ}$ converge to $f (x)$ and $g (x)$ , respectively, on $ℐ$ ; then

\sum_{κ = 0}^{\infty} (𝒸_{κ} \pm 𝒹_{κ}) x^{κ}

converges to

f (x) \pm g (x)

on $ℐ$ .

Operations

Part II

Let $\sum_{κ = 0}^{\infty} 𝒸_{κ} x^{κ}$ converge to $f (x)$ on $ℐ$ , then for $m \in ℕ$ and $b \in ℝ$

\sum_{κ = 0}^{\infty} b x^{m} 𝒸_{κ} x^{κ}

converges to

b x^{m} f (x)

on $ℐ$ .

Operations

Part III

Let $\sum_{κ = 0}^{\infty} 𝒸_{κ} x^{κ}$ converge to $f (x)$ , on $ℐ$ , then for $m \in ℕ$ and $b \in ℝ$

\sum_{κ = 0}^{\infty} 𝒸_{κ} {(b x^{m})}^{κ}

converges to

f (b x^{m})

for all $x$ such that $b x^{m} \in ℐ$ .

Multiplication

\begin{matrix} (\sum_{κ = 0}^{\infty} 𝒸_{κ} x^{κ}) (\sum_{κ = 0}^{\infty} 𝒹_{κ} x^{κ}) & = & (𝒸_{0} + 𝒸_{1} x + 𝒸_{2} x^{2} + 𝒸_{3} x^{3} + \dots) \\ (𝒹_{0} + 𝒹_{1} x + 𝒹_{2} x^{2} + 𝒹_{3} x^{3} + \dots) \\ = & 𝒸_{0} 𝒹_{0} \\ + (𝒸_{0} 𝒹_{1} + 𝒸_{1} 𝒹_{0}) x \\ + (𝒸_{0} 𝒹_{2} + 𝒸_{1} 𝒹_{1} + 𝒸_{2} 𝒹_{0}) x^{2} \\ + (𝒸_{0} 𝒹_{3} + 𝒸_{1} 𝒹_{2} + 𝒸_{2} 𝒹_{1} + 𝒸_{3} 𝒹_{0}) x^{3} \\ + \dots \end{matrix}

Differentiation

Let $\sum_{κ = 0}^{\infty} 𝒸_{κ} {(x - 𝒶)}^{κ}$ be convergent on $ℐ = (𝒶 - 𝓇, 𝒶 + 𝓇)$ and

\begin{matrix} f (x) & = & \sum_{κ = 0}^{\infty} 𝒸_{κ} {(x - 𝒶)}^{κ} \\ = & 𝒸_{0} + 𝒸_{1} (x - 𝒶) + 𝒸_{2} {(x - 𝒶)}^{2} + 𝒸_{3} {(x - 𝒶)}^{3} + \dots \end{matrix}

Then on $ℐ$

\begin{matrix} f^{'} (x) & = & \sum_{κ = 1}^{\infty} κ 𝒸_{κ} {(x - 𝒶)}^{κ - 1} \\ = & 𝒸_{1} + 2 𝒸_{2} (x - 𝒶) + 3 𝒸_{3} {(x - 𝒶)}^{2} + 4 𝒸_{4} {(x - 𝒶)}^{3} + \dots \end{matrix}

Integration

Let $\sum_{κ = 0}^{\infty} 𝒸_{κ} {(x - 𝒶)}^{κ}$ be convergent on $ℐ = (𝒶 - 𝓇, 𝒶 + 𝓇)$ and

\begin{matrix} f (x) & = & \sum_{κ = 0}^{\infty} 𝒸_{κ} {(x - 𝒶)}^{κ} \\ = & 𝒸_{0} + 𝒸_{1} (x - 𝒶) + 𝒸_{2} {(x - 𝒶)}^{2} + 𝒸_{3} {(x - 𝒶)}^{3} + \dots \end{matrix}

Then on $ℐ$

\begin{matrix} \int f (x) ⅆ x & = & C + \sum_{κ = 0}^{\infty} 𝒸_{κ} \frac{{(x - 𝒶)}^{κ + 1}}{κ + 1} \\ = & C + 𝒸_{0} (x - 𝒶) + 𝒸_{1} \frac{{(x - 𝒶)}^{2}}{2} + 𝒸_{2} \frac{{(x - 𝒶)}^{3}}{3} + \dots \end{matrix}

Taylor expansion

Term-by-term

\begin{matrix} f (x) & = & \sum_{κ = 0}^{\infty} 𝒸_{κ} {(x - 𝒶)}^{κ} = 𝒸_{0} + 𝒸_{1} (x - 𝒶) + 𝒸_{2} {(x - 𝒶)}^{2} + \dots \\ f^{'} (x) & = & \sum_{κ = 1}^{\infty} κ 𝒸_{κ} {(x - 𝒶)}^{κ - 1} = 𝒸_{1} + 2 𝒸_{2} (x - 𝒶) + 3 𝒸_{3} {(x - 𝒶)}^{2} + \dots \\ f^{″} (x) & = & \sum_{κ = 2}^{\infty} (κ - 1) κ 𝒸_{κ} {(x - 𝒶)}^{κ - 1} = 2 𝒸_{2} + 2 \cdot 3 \cdot 𝒸_{3} (x - 𝒶) + \dots \\ f^{‴} (x) & = & \sum_{κ = 3}^{\infty} (κ - 2) (κ - 1) κ 𝒸_{κ} {(x - 𝒶)}^{κ - 3} = 6 𝒸_{3} + \dots \\ ⋮ \\ f^{(n)} (x) & = & \sum_{κ = n}^{\infty} \frac{κ!}{(κ - n)!} 𝒸_{κ} {(x - 𝒶)}^{κ - n} = n! 𝒸_{n} + \frac{(n + 1)!}{1!} 𝒸_{n + 1} (x - 𝒶) + \dots \end{matrix}

Taylor series

Let $f (x)$ have derivatives of all orders on $(𝒶 - 𝓇, 𝒶 + 𝓇)$ . The Taylor series generated by $f (x)$ at $𝒶$ is

\sum_{κ = 0}^{\infty} \frac{f^{(κ)} (𝒶)}{κ!} {(x - 𝒶)}^{κ} = f (𝒶) + \frac{f^{'} (𝒶)}{1!} (x - 𝒶) + \frac{f^{″} (𝒶)}{2!} {(x - 𝒶)}^{2} + \dots

The Maclaurin series of $f (x)$ is the Taylor series at zero.

Uniqueness

If $f (x)$ has a power series at $𝒶$ that converges to $f (x)$ on $(𝒶 - 𝓇, 𝒶 + 𝓇)$ , then the power series is the Taylor series of $f (x)$ at $𝒶$ .

Taylor polynomial

If $f (x)$ has $M$ 'th order derivative on an open interval that contains $𝒶$ then for any $n = 0, 1, \dots, M$ , the Taylor polynomial of order $n$ generated by $f (x)$ at $𝒶$ is

p_{n} (x) = f (𝒶) + f^{'} (𝒶) (x - 𝒶) + \frac{f^{″} (𝒶)}{2!} {(x - 𝒶)}^{2} + \dots + \frac{f^{(n)} (𝒶)}{n!} {(x - 𝒶)}^{n}

Taylor approximation

Let $f (x)$ be defined on an open interval $ℐ$ that contains $𝒶$ and $f^{(n + 1)} (x)$ exist on $ℐ$ , then for every $x \in ℐ$ there is $𝒸 \in (𝒶, x)$ such that

f (x) = \sum_{κ = 0}^{n} \frac{f^{(κ)} (𝒶)}{κ!} {(x - 𝒶)}^{κ} + \frac{f^{(n + 1)} (𝒸)}{(n + 1)!} {(x - 𝒶)}^{n + 1}

Series

Examples

Geometric series

\sum_{κ = 1}^{\infty} α 𝓇^{κ}