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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
    \({\large\int\limits_0^x\normalsize} {f\left( t \right)dt} =\) \({\large\int\limits_0^x\normalsize} {{a_0}dt} + {\large\int\limits_0^x\normalsize} {{a_1}tdt} \) \(+\;{\large\int\limits_0^x\normalsize} {{a_2}{t^2}dt} + \ldots\) \(+\;{\large\int\limits_0^x\normalsize} {{a_n}{t^n}dt} + \ldots =\) \({a_0}x + {a_1}{\large\frac{{{x^2}}}{2}\normalsize} + {a_2}{\large\frac{{{x^3}}}{3}\normalsize} + \ldots =\) \( \sum\limits_{n = 0}^\infty {{a_n}{\large\frac{{{x^{n + 1}}}}{{n + 1}}\normalsize}} + C.\)