Question

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

Short answer

The power series representation for the function \(f(x) = \sum\limits_{n = 0}^\infty {{{( - 1)}^n}} {x^n}\) and the interval of convergence is \(( - 1,1)\)

Step-by-step solution

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

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

Express the given function \(f(x) = \frac{1}{{1 + x}}\) as follows,

\(\frac{1}{{1 + x}} = \frac{1}{{1 - ( - x)}}\)

Compare to equation \((1),f(x)\) is a sum of a geometric series with \(a = 1\) and \(r = - x\).

Then,

\(\begin{aligned}f(x) &= \sum\limits_{n = 0}^\infty a {r^n}\\f(x) &= \sum\limits_{n = 0}^\infty {(1)} {( - x)^n}\\f(x) &= \sum\limits_{n = 0}^\infty {{{( - 1)}^n}} {x^n}\end{aligned}\)

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

Calculation for the interval of convergence

Since the geometric series converges if \(|r| < 1,| - x| < 1\).

\(|x| < 1\quad (|x| = | - x|)\)

\( - 1 < x < 1\)

The interval of convergence is \(( - 1,1)\) and radius of convergence is \(R = 1\).