Power Series as a Function

Since power series are defined relevant to a value $x$. We can define some properties for two series $f(x)=\sum a_n{x}^n$ for $|x|<R_1$ and $g(x)=\sum b_n{x}^n$ for $|x|<R_2$:

• $c{x}^k\sum f(x)=\sum c{a}_n{x}^{n+k}$ $|x|<R_1$
• $f(c{x}^k)=\sum a_n{c}^n{x}^{nk}$ $|c{x}^k|<R_1$
• $f(x-c)=\sum a_n(x-c{)}^n$ $|x-c|<R_1$
• $f(x)+g(x)=\sum (a_n+b_n)x^n$ $|x|<min(R_1,R_2)$

It is also possible to differentiate power series

$$f(x)=\sum c_n(x-a{)}^n=c_0+c_1(x-a)+c_2(x-a{)}^2...$$ $$f^{\prime }(x)=c_1+2c_2(x-a)+3c_3(x-a{)}^2=\sum n{c}_n(x-a{)}^{n-1}$$ $$\int f(x)dx=c+c_0(x-a)+\frac{c_1(x-a{)}^2}{2}=\sum \frac{c_n(x-a{)}^{n+1}}{n+1}$$

The radius of convergence remains the same for both integration and differentiation