# Differentiation and Integration of Power Series

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 or tap a problem to see the solution.

### Example 1

Show that
${\frac{1}{{1 + x}} = 1 – x + {x^2} }-{ {x^3} + {x^4} – \ldots } = {\sum\limits_{n = 0}^\infty {{a_n}{x^n}}}$
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
${\frac{1}{{1 + x}} = 1 – x + {x^2} }-{ {x^3} + {x^4} – \ldots } = {\sum\limits_{n = 0}^\infty {{a_n}{x^n}}}$
for $$\left| {x} \right| \lt 1.$$

Solution.

First we consider the power series:

${1 + x + {x^2} }+{ {x^3} + \ldots }$

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

$\frac{1}{{2 – x}} = \frac{{\frac{1}{2}}}{{1 – \frac{x}{2}}}.$

As you can see, 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 $$\left| x \right| \lt 2.$$

Page 1
Problems 1-2
Page 2
Problems 3-7