# Properties of Power Series

• Functional series: $$\sum\limits_{n = 0}^\infty {{u_n}\left( x \right)}$$
Functions: $$f\left( x \right),$$ $${u_0}\left( x \right),$$ $${u_1}\left( x \right), \ldots,$$ $${u_n}\left( x \right)$$
Power series: $$\sum\limits_{n = 0}^\infty {{a_n}{x^n}} ,$$ $$\sum\limits_{n = 0}^\infty {{a_n}{{\left( {x – {x_0}} \right)}^n}}$$
Coefficients of a power series: $${a_0},{a_1}, \ldots ,{a_n}$$
Radius of convergence: $$R$$
Real numbers: $$x$$, $${x_0}$$
Whole numbers: $$n$$
1. A series whose terms are functions is called a functional series. In general, a functional series can be written in the form
$$\sum\limits_{n = 0}^\infty {{u_n}\left( x \right)} =$$ $${u_0}\left( x \right) + {u_1}\left( x \right)$$ $$+\; {u_2}\left( x \right) + \ldots$$ $$+\; {u_n}\left( x \right) + \ldots,$$
where $${u_i}\left( x \right)$$ are functions of the variable $$x$$.
2. A functional series, the terms of which are power functions of the variable $$x$$, is called a power series:
$$\sum\limits_{n = 0}^\infty {{a_n}{x^n}} =$$ $${a_0} + {a_1}x$$ $$+\; {a_2}{x^2} + \ldots$$ $$+\; {a_n}{x^n} + \ldots ,$$
where $${a_i}$$ are the coefficients of the power series (constant real numbers).
3. A power series in powers of $${\left( {x – {x_0}} \right)}$$ is also often considered:
$$\sum\limits_{n = 0}^\infty {{a_n}{{\left( {x – {x_0}} \right)}^n}} =$$ $${a_0} + {a_1}\left( {x – {x_0}} \right)$$ $$+\; {a_2}{\left( {x – {x_0}} \right)^2} + \ldots$$ $$+\; {a_n}{\left( {x – {x_0}} \right)^n} + \ldots ,$$
where the point $${x_0}$$ is called the center of the power series.
4. Interval of convergence
Consider a function $$f\left( x \right) =$$ $$\sum\limits_{n = 0}^\infty {{a_n}{{\left( {x – {x_0}} \right)}^n}}.$$ The domain of this function is the set of those values of $$x$$ for which the series is convergent. The domain of such function is called the interval of convergence.
5. Radius of convergence
If the interval of convergence of a power series is represented in the form $$\left( {{x_0} – R,{x_0} + R} \right)$$, where $$R \gt 0$$, then the value of $$R$$ is called the radius of convergence. The power series converges absolutely at every point of the interval of convergence. Convergence at the endpoints $${{x_0} – R}$$ and $${{x_0} + R}$$ must be examined separately.
6. Radius of convergence by the ratio test
$$R = \lim\limits_{n \to \infty } \left| {\large\frac{{{a_n}}}{{{a_{n + 1}}}}\normalsize} \right|$$
7. Radius of convergence by the root test
$$R = \lim\limits_{n \to \infty } {\large\frac{1}{{\sqrt[n]{{{a_n}}}}\normalsize}}$$
8. Differentiation of power series
Consider a power series
$$f\left( x \right) = \sum\limits_{n = 0}^\infty {{a_n}{x^n}} =$$ $${a_0} + {a_1}x$$ $$+\; {a_2}{x^2} + \ldots ,$$
with the radius of convergence $$R \gt 0$$. The function $$f\left( x \right) = \sum\limits_{n = 0}^\infty {{a_n}{x^n}}$$ is continuous for $$\left| x \right| \lt R$$. The power series can be differentiated term-by-term inside the interval of convergence. The derivative of the power series is given by the formula
$$f’\left( x \right) =$$ $${\large\frac{d}{{dx}}\normalsize}{a_0} + {\large\frac{d}{{dx}}\normalsize}{a_1}x$$ $$+\;{\large\frac{d}{{dx}}\normalsize}{a_2}{x^2} + \ldots =$$ $${a_1} + 2{a_2}x + 3{a_3}{x^2} + \ldots =$$ $$\sum\limits_{n = 1}^\infty {n{a_n}{x^{n – 1}}} .$$
9. Integration of power series
A power series can be also integrated term-by-term on an interval lying inside the interval of convergence. If $$-R \lt b \lt x \lt R,$$ then the following expression is valid:
$${\large\int\limits_b^x\normalsize} {f\left( t \right)dt} =$$ $${\large\int\limits_b^x\normalsize} {{a_0}dt} + {\large\int\limits_b^x\normalsize} {{a_1}tdt}$$ $$+\;{\large\int\limits_b^x\normalsize} {{a_2}{t^2}dt} + \ldots$$ $$+\;{\large\int\limits_b^x\normalsize} {{a_n}{t^n}dt} + \ldots$$
If the series is integrated on the interval $$\left[ {0,x} \right],$$ then the integral is given by the formula
$${\large\int\limits_0^x\normalsize} {f\left( t \right)dt} =$$ $${\large\int\limits_0^x\normalsize} {{a_0}dt} + {\large\int\limits_0^x\normalsize} {{a_1}tdt}$$ $$+\;{\large\int\limits_0^x\normalsize} {{a_2}{t^2}dt} + \ldots$$ $$+\;{\large\int\limits_0^x\normalsize} {{a_n}{t^n}dt} + \ldots =$$ $${a_0}x + {a_1}{\large\frac{{{x^2}}}{2}\normalsize} + {a_2}{\large\frac{{{x^3}}}{3}\normalsize} + \ldots =$$ $$\sum\limits_{n = 0}^\infty {{a_n}{\large\frac{{{x^{n + 1}}}}{{n + 1}}\normalsize}} + C.$$