Question

Find a power series representation for the function. (give your power series representation centered at x=0)

f(x)=x/14x²+1
[infinity]
f(x)=∑ (___)
n=0

Answer 1

The power series representation of the function f(x) = x/(14x² + 1), centered at x=0, is obtained by employing the geometric series formula and is Σ (n=0) (-14^n x^(2n+1)).

Step-by-step explanation:

To find a power series representation of the function f(x) = x/(14x² + 1), we need to manipulate the function into a form that allows us to use a known series expansion. First, let's consider the geometric series, a / (1 - r), which converges to a + ar + ar² + ... Provided that |r| < 1. In our case, we need to express f(x) in a related form:

f(x) = x / (1 - (-14x²))

We can now write the geometric series for f(x) as a power series:

f(x) = x(1 + (-14x²) + (-14x²)² + ... + (-14x²)^n + ...)

Expressed in sigma notation, the power series representation centered at x=0 is:

infinityf(x) = Σ (n=0) (-14^n x^(2n+1))