Question

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

Answer 1

`f(x) = \sum\limits^(\infty)_(n=0) (-(1)^n\cdot x^n)/(5^(n+1))`

Explanation:

Given

`f(x) = (1)/(5 + x)`

Required

The power series centered at
`x = 0`

We have:

`f(x) = (1)/(5 + x)`

Factor out 5 from the denominator

`f(x) = (1)/(5(1 + (x)/(5)))`

Rewrite as:

`f(x) = ((1)/(5))/((1 + (x)/(5)))`

Further, rewrite as:

`f(x) = (1)/(5)(1 + (x)/(5))^(-1)`

Expand the bracket

`f(x) = (1)/(5)(1 - (x)/(5) + ((x)/(5))^2 - ((x)/(5))^3+..........)`

Evaluate all exponents

`f(x) = (1)/(5)(1 - (x)/(5) + (x^2)/(25) - (x^3)/(125)+......)`

Open brackets

`f(x) = (1)/(5) - (x)/(5^2) + (x^2)/(5^3) - (x^3)/(5^4)+......`

Notice the pattern as:

`f(x) = (1)/(5) - (x)/(5^2) + (x^2)/(5^3) - (x^3)/(5^4)+......\± (x^n)/(5^(n+1))`

So, the power series is:

`f(x) = \sum\limits^(\infty)_(n=0) (-(1)^n\cdot x^n)/(5^(n+1))`