Question

To determine the power series representation for the function \(f(x) = \frac{1}{{x + 10}}\) and determine the interval of convergence.

Short answer

The power series representation for the given function is \(f(x) = \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}{x^n}}}{{{{10}^{n + 1}}}}} \) and the interval of convergence is \(\left( { - \frac{3}{2},\frac{3}{2}} \right)\).

Step-by-step solution

Concept of Geometric Series

Geometric Series

The sum of the geometric series with initial term\(a\)and common ratio\(r\)is\(\sum\limits_{n = 0}^\infty a {r^n} = \frac{a}{{1 - r}}\)

Calculation of the expression\(f(x) = \frac{1}{{x + 10}}\)

Consider,

\(f(x) = \frac{1}{{x + 10}}\).

Divide the numerator and the denominator by \(10\)

\(\begin{aligned}f(x) &= \frac{{\frac{1}{{10}}}}{{\frac{x}{{10}} + \frac{{10}}{{10}}}}\\f(x) &= \frac{{\frac{1}{{10}}}}{{\frac{x}{{10}} + 1}}\end{aligned}\)

Express the function as follows,

\(\frac{{\frac{1}{{10}}}}{{\frac{x}{{10}} + 1}} = \frac{{\frac{1}{{10}}}}{{1 - \left( { - \frac{x}{{10}}} \right)}}\)

Compare the equation with geometric series as,\(f(x)\)is a sum of a geometric series with \(a = \frac{1}{{10}}\) and \(r = - \frac{x}{{10}}\).

Therefore,

\(\frac{{\frac{1}{{10}}}}{{1 - \left( { - \frac{x}{{10}}} \right)}} = \sum\limits_{n = 0}^\infty {\frac{1}{{10}}} {\left( { - \frac{x}{{10}}} \right)^n}\).

Simplification of the expression\(\sum\limits_{n = 0}^\infty   {\frac{1}{{10}}} {\left( { - \frac{x}{{10}}} \right)^n}\)

Simplify the terms further as shown below,

\(\begin{aligned}\sum\limits_{n = 0}^\infty {\frac{1}{{10}}} {\left( { - \frac{x}{{10}}} \right)^n} &= \sum\limits_{n = 0}^\infty {\frac{1}{{10}}} {\left( { - \frac{1}{{10}}} \right)^n}{x^n}\\\sum\limits_{n = 0}^\infty {\frac{1}{{10}}} {\left( { - \frac{x}{{10}}} \right)^n} &= \sum\limits_{n = 0}^\infty {{{( - 1)}^n}} \frac{1}{{10}}{\left( {\frac{1}{{10}}} \right)^n}{x^n}\\\sum\limits_{n = 0}^\infty {\frac{1}{{10}}} {\left( { - \frac{x}{{10}}} \right)^n} &= \sum\limits_{n = 0}^\infty {{{( - 1)}^n}} {\left( {\frac{1}{{10}}} \right)^{n + 1}}{x^n}\\\sum\limits_{n = 0}^\infty {\frac{1}{{10}}} {\left( { - \frac{x}{{10}}} \right)^n} &= \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}{x^n}}}{{{{10}^{n + 1}}}}} \end{aligned}\)

Thus, the power series representation of the function is \(f(x) = \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}{x^n}}}{{{{10}^{n + 1}}}}} \).

Calculation for the interval of convergence

We know that the geometric series converges when \(|r| < 1,\left| { - \frac{x}{{10}}} \right| < 1\)

\(\frac{1}{{10}}|x| < 1\)

\(|x| < 10\)

\( - 10 < x < 10\)

Therefore, the interval of convergence is \(( - 10,10)\).