9. Power Series

Centred about 0

f (x) = \sum_{n = 0}^{\infty} a_{n} x^{n}

Thm Convergence and Absolute Convergence

If $\sum_{}^{} a_{n} x^{n}$ converges at $x_{0} \in R$ then it converges absolutely for every $| x | < | x_{0} |$

Proof

If $\sum_{}^{} a_{n} x^{n}$ converges then $(a_{n} x^{n}) \to 0$ and the sequence is bounded

Let $M > 0$ st $| a_{n} x^{n} | < M$ for all $n \in N$

If $x \in R$ with $| x | < | x_{0} |$ , $| a_{n} x^{n} | = | a_{n} x_{0}^{n} | | \frac{x^{n}}{x_{0}^{n}} | \leq M {| \frac{x}{x_{0}} |}^{n}$

M \sum_{}^{} {| \frac{x}{x_{0}} |}^{n}

is a geometric series with $| \frac{x}{x_{0}} | < 1 ⟹$ converges
by comparison test $\sum_{}^{} | a_{n} x^{n} |$ converges 3. Series

Def Radius of Convergence

The radius of convergence of $\sum_{}^{} a_{n} x^{n}$

R = {\begin{cases} 0 & if the limsup | a_{n} |^{1 / n} \to + \infty \\ + \infty & if limsup | a_{n} |^{1 / n} \to 0 \\ \frac{1}{limsup | a_{n} |^{1 / n}} & otherwise \end{cases}

$\sum_{}^{} a_{n} x^{n}$ converges on $(- R, R)$

If it exists $R = lim | \frac{a_{n}}{a_{n + 1}} |$

Thm Absolute then Uniform

If a power series $\sum_{n = 0}^{\infty} a_{n} x^{n}$ converges absolutely at a point $x_{0}$ then it converges uniformly on the closed interval $[- c, c]$ where $c = | x_{0} |$

Proof

If $| p | < | q | ⟹ | p^{n} | < | q^{n} |$ so using the Weirerstrass M-Test with $M_{n} = | a_{n} x^{n} |$

Thm Abel's Lemma

Let $b_{1} \geq b_{2} \geq b_{3} \geq \dots \geq 0$ and let $\sum_{n = 1}^{\infty} a_{n}$ be a series for which the partial sums are bounded.

| a_{1} + a_{2} + \cdot \cdot \cdot + a_{n} | \leq A

for all $n \in N$ then for all $n \in N$

| a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3} + \cdot \cdot \cdot + a_{n} b_{n} | \leq A b_{1} .

Thm Abel's Theorem

Let $g (x) = \sum_{}^{} a_{n} x^{n}$ converges at $x = R > 0$ then $g$ converges uniformly $[0, R]$

Thm Uniform Compact

If a power series converges on $A \subseteq R$ then in converges uniformly on any compact subset $K \subseteq A$
This fact leads to the desirable conclusion that a power series is continuous at every point at which it converges.

Thm Differentiated Convergence

If $\sum_{n = 0}^{\infty} a_{n} x^{n}$ converges for all $x \in (- R, R)$
Then $\sum_{n = 1}^{\infty} n \cdot a_{n} x^{n - 1}$ converges at each $x \in (- R, R)$ and consequently the convergence is uniform $K \subset (- R, R)$ if $K$ is compact.

Also if

f (x) = \sum_{n = 0}^{\infty} a_{n} x^{n}

converges on an interval $A \subseteq R$ . The function is continuous on $A$ and differentiable on any open interval $(- R, R) \subseteq A$ . Where the derivative is:

f^{'} (x) = \sum_{n = 1}^{\infty} n \cdot a_{n} x^{n - 1}

and it is infinitely differentiable on $(- R, R)$