Properties of Power Series

Now we quickly state some of the important properties of the power series. Consider two power series

$$\sum_{n=0}^\infty a_n (x- x_0)^n \;\; {\mbox{ and }} \; \; \sum_{n=0}^\infty b_n (x- x_0)^n$$

with radius of convergence $R_1>0$ and $R_2 > 0,$ respectively. Let $F(x)$ and $G(x)$ be the functions defined by the two power series defined for all $x \in I,$ where $I = (-R + x_0, x_0 + R)$ with $R = \min \{ R_1, R_2 \}.$ Note that both the power series converge for all $x \in I.$

With $F(x), \;G(x)$ and $I$ as defined above, we have the following properties of the power series.

1. EQUALITY OF POWER SERIES
   The two power series defined by $F(x)$ and $G(x)$ are equal for all $x \in I$ if and only if
   $$a_n = b_n \; {\mbox{ for all }} n = 0, 1, 2, \ldots.$$
   In particular, if $\sum\limits_{n=0}^\infty a_n (x-x_0)^n = 0$ for all $x \in I,$ then
   $$a_n = 0 \; {\mbox{ for all }} \; n=0,1,2,\ldots.$$

2. TERM BY TERM ADDITION
   For all $x \in I,$ we have
   $$F(x) + G(x) = \sum_{n=0}^\infty (a_n + b_n) (x-x_0)^n$$
   Essentially, it says that in the common part of the regions of convergence, the two power series can be added term by term.
3. MULTIPLICATION OF POWER SERIES
   Let us define
   $$c_0 = a_0 b_0, \; {\mbox{ and inductively }} \; c_n = \sum_{j=1}^n a_{n-j} b_j.$$
   Then for all $x \in I,$ the product of $F(x)$ and $G(x)$ is defined by
   $$H(x) = F(x) G(x) = \sum\limits_{n=0}^\infty c_n (x-x_0)^n.$$
   $H(x)$ is called the ``Cauchy Product" of $F(x)$ and $G(x).$

   Note that for any $n \geq o,$ the coefficient of $x^n$ in

   $$\left(\sum\limits_{j=0}^\infty a_j (x-x_0)^j \right) \cdot \left... ...x_0)^k \right) \; \; {\mbox{ is }} \;\; c_n = \sum\limits_{j=1}^n a_{n-j} b_j.$$

4. TERM BY TERM DIFFERENTIATION
   The term by term differentiation of the power series function $F(x)$ is
   $$\sum_{n=1}^\infty n a_n (x - x_0)^n.$$
   Note that it also has $R_1$ as the radius of convergence as by Theorem 9.1.4 $\lim\limits_{n \longrightarrow \infty} \sqrt[n]{\vert a_n\vert} = \cdot \frac{1}{R_1}$ and
   $$\lim_{n \longrightarrow \infty} \sqrt[n]{\vert n a_n\vert} = \li... ...im_{n \longrightarrow \infty} \sqrt[n]{\vert a_n\vert} = 1 \cdot \frac{1}{R_1}.$$
   Let $0 < r < R_1.$ Then for all $x \in (-r + x_0, x_0 + r),$ we have
   $$\frac{d}{dx} F(x) = F^\prime(x) = \sum_{n=1}^\infty n a_n (x - x_0)^n.$$
   In other words, inside the region of convergence, the power series can be differentiated term by term.

In the following, we shall consider power series with $x_0 = 0$ as the centre. Note that by a transformation of $X = x - x_0,$ the centre of the power series can be shifted to the origin.

EXERCISE 9.1.1  

1. which of the following represents a power series (with centre $x_0$ indicated in the brackets) in $x?$
   1. $1 + x^2 + x^4 + \cdots + x^{2n} + \cdots \hspace{1.2in} (x_0 = 0).$
   2. $1 + \sin x + (\sin x)^2 + \cdots + (\sin x)^n + \cdots \hspace{.52in} (x_0 = 0).$
   3. $1 + x \vert x\vert + x^2 \vert x^2\vert + \cdots + x^n \vert x^n\vert + \cdots \hspace{.73in} (x_0 = 0).$
2. Let $f(x)$ and $g(x)$ be two power series around $x_0 = 0,$ defined by
   

   $$f(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots + (-1)^n \frac{x^{2n+1}}{(2n+1)!} + \cdots$$

   $${\mbox{ and }} \;\; g(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots + (-1)^n \frac{x^{2n}}{(2n)!} + \cdots.$$

   
   Find the radius of convergence of $f(x)$ and $g(x).$ Also, for each $x$ in the domain of convergence, show that
   $$f'(x) = g(x) \;\; {\mbox{ and }} \;\; g'(x) = -f(x).$$
   [Hint: Use Properties $1, 2, 3$ and $4$ mentioned above. Also, note that we usually call $f(x)$ by $\sin x$ and $g(x)$ by $\cos x.$ ]
3. Find the radius of convergence of the following series centred at $x_0 = -1.$
   1. $1 + (x+1) + \frac{(x+1)^2}{2!} + \cdots + \frac{(x+1)^n}{n!} + \cdots .$
   2. $1 + (x+1) + 2 (x+1)^2 + \cdots + n (x+1)^n + \cdots .$