Calculus
Infinite Sequences and Series
Differentiation and Integration of Power Series
Problems 1-2
Problems 3-7
Let the power series \(\sum\limits_{n = 0}^\infty {{a_n}{x^n}}\) have the radius of convergence \(R \gt 0.\) Let
{f\left( x \right) = \sum\limits_{n = 0}^\infty {{a_n}{x^n}} }
= {{a_0} + {a_1}x }+{ {a_2}{x^2} + \ldots ,\;\;}\kern-0.3pt{\left| x \right| \lt R.}
\]
Then for \(\left| x \right| \lt R\) the function \(f\left( x \right) = \sum\limits_{n = 0}^\infty {{a_n}{x^n}} \) is continuous. The power series can be differentiated term-by-term inside the interval of convergence. The derivative of the power series exists and is given by the formula
{f’\left( x \right) }
= {\frac{d}{{dx}}{a_0} + \frac{d}{{dx}}{a_1}x }+{ \frac{d}{{dx}}{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}}} .}
\]
The power series can be also integrated term-by-term on an interval lying inside the interval of convergence. Hence, if \( – R \lt b \lt x \lt R,\) the following expression is valid:
{\int\limits_b^x {f\left( t \right)dt} }
= {\int\limits_b^x {{a_0}dt} + \int\limits_b^x {{a_1}tdt} }
+ {\int\limits_b^x {{a_2}{t^2}dt} + \ldots }
+ {\int\limits_b^x {{a_n}{t^n}dt} + \ldots }
\]
If the series is integrated on the interval \(\left[ {0,x} \right],\) we can write:
{\int\limits_0^x {f\left( t \right)dt} }
= {\int\limits_0^x {{a_0}dt} + \int\limits_0^x {{a_1}tdt} }
+ {\int\limits_0^x {{a_2}{t^2}dt} + \ldots }
+ {\int\limits_0^x {{a_n}{t^n}dt} + \ldots }
= {{a_0}x + {a_1}\frac{{{x^2}}}{2} + {a_2}\frac{{{x^3}}}{3} + \ldots }
= {\sum\limits_{n = 0}^\infty {{a_n}\frac{{{x^{n + 1}}}}{{n + 1}}} }+{ C.}
\]
Solved Problems
Click on problem description to see solution.
✓ Example 1
Show that
{\frac{1}{{1 + x}} = 1 – x + {x^2} }-{ {x^3} + {x^4} – \ldots }
= {\sum\limits_{n = 0}^\infty {{a_n}{x^n}} \;\;}\kern-0.3pt{\text{for}\;\;\left| x \right| \lt 1.}
\]
✓ Example 2
Find a power series for the rational fraction \(\large\frac{1}{{2 – x}}\normalsize.\)
✓ Example 3
Find a power series for \(\large\frac{{6x}}{{5{x^2} – 4x – 1}}\normalsize.\)
✓ Example 4
Find a power series representation for the function \(\ln \left( {1 + x} \right),\;\left| x \right| \lt 1.\)
✓ Example 5
Represent the integral \(\int\limits_0^x {{\large\frac{{\ln \left( {1 + t} \right)}}{t}\normalsize} dt}\) as a power series expansion.
✓ Example 6
Obtain a power series representation for the exponential function \({e^x}.\)
✓ Example 7
Find a power series expansion for the hyperbolic sine function \(\sinh x.\)
Example 1.
Show that
{\frac{1}{{1 + x}} = 1 – x + {x^2} }-{ {x^3} + {x^4} – \ldots }
= {\sum\limits_{n = 0}^\infty {{a_n}{x^n}} \;\;}\kern-0.3pt{\text{for}\;\;\left| x \right| \lt 1.}
\]
Solution.
First we consider the power series:
This is a geometric series with ratio \(x.\) Therefore, it converges for \(\left| x \right| \lt 1.\) The sum of the series is \({\large\frac{1}{{1 – x}}\normalsize}.\) Substituting \(-x\) for \(x,\) we have
{1 – x + {x^2} }-{ {x^3} + \ldots }
= {\frac{1}{{1 – \left( { – x} \right)}} }
= {\frac{1}{{1 + x}}\;\;}\kern-0.3pt{\text{for}\;\;\left| x \right| \lt 1.}
\]
Thus,
{\frac{1}{{1 + x}} = 1 – x + {x^2} }-{ {x^3} + {x^4} – \ldots }
= {\sum\limits_{n = 0}^\infty {{{\left( { – 1} \right)}^n}{x^n}} \;\;}\kern-0.3pt
{\text{for}\;\;\left| x \right| \lt 1.}
\]
Example 2.
Find a power series for the rational fraction \(\large\frac{1}{{2 – x}}\normalsize.\)
Solution.
We can write this function as
As can be seen, this is the sum of the infinite geometric series with the first term \({\large\frac{1}{2}\normalsize}\) and ratio \({\large\frac{x}{2}\normalsize}:\)
{\frac{1}{2} + \frac{1}{2}\frac{x}{2} + \frac{1}{2}{\left( {\frac{x}{2}} \right)^2} }
+ {\frac{1}{2}{\left( {\frac{x}{2}} \right)^3} + \ldots }
= {\frac{1}{2} + \frac{x}{{{2^2}}} + \frac{{{x^2}}}{{{2^3}}} }+{ \frac{{{x^3}}}{{{2^4}}} + \ldots }
= {\sum\limits_{n = 0}^\infty {\frac{{{x^n}}}{{{2^{n + 1}}}}} .}
\]
The given power series converges for \(\left| x \right| \lt 2.\)
Problems 1-2
Problems 3-7