Question

Find a power series representation for the function. (Give your power series representation centered at x=0 .) f(x)=15+x

Answer 1

Answer:

$f(x) = \sum\limits^{\infty}_{n=0} \frac{-(1)^n\cdot x^n}{5^{n+1}}$

Step-by-step explanation:

Given

$f(x) = \frac{1}{5 + x}$

Required

The power series centered at $x = 0$

We have:

$f(x) = \frac{1}{5 + x}$

Factor out 5 from the denominator

$f(x) = \frac{1}{5(1 + \frac{x}{5})}$

Rewrite as:

$f(x) = \frac{\frac{1}{5}}{(1 + \frac{x}{5})}$

Further, rewrite as:

$f(x) = \frac{1}{5}(1 + \frac{x}{5})^{-1}$

Expand the bracket

$f(x) = \frac{1}{5}(1 - \frac{x}{5} + (\frac{x}{5})^2 - (\frac{x}{5})^3+..........)$

Evaluate all exponents

$f(x) = \frac{1}{5}(1 - \frac{x}{5} + \frac{x^2}{25} - \frac{x^3}{125}+......)$

Open brackets

$f(x) = \frac{1}{5} - \frac{x}{5^2} + \frac{x^2}{5^3} - \frac{x^3}{5^4}+......$

Notice the pattern as:

$f(x) = \frac{1}{5} - \frac{x}{5^2} + \frac{x^2}{5^3} - \frac{x^3}{5^4}+......\± \frac{x^n}{5^{n+1}}$

So, the power series is:

$f(x) = \sum\limits^{\infty}_{n=0} \frac{-(1)^n\cdot x^n}{5^{n+1}}$