Final answer:
The power series representation of the function f(x) is Σ(-1)^n * x^(2n+1) / 5^(n+1), and the radius of convergence (R) is √5. The coefficient and the exponent in the series expansion are adjusted in accordance with the geometric series formula and the function structure.
Step-by-step explanation:
A student has asked for the power series representation of the function f(x) = x/(5 + x^2) and the determination of the radius of convergence of that series. To find this power series representation, we can use the geometric series formula which is:
1/(1 - r) = Σr^n from n=0 to n=∞, where |r| < 1.
First, we rewrite the function in a form that allows us to use the geometric series:
f(x) = x/5 * 1/(1 + (x^2/5)) = (1/5)x * Σ(-1)^n * (x^2/5)^n
Now, we can write the power series expansion by distributing x and simplifying:
f(x) = Σ(-1)^n * x^(2n+1) / 5^(n+1), for |x^2/5| < 1
The radius of convergence (R) can be found by the inequality |x^2/5| < 1. Solving for |x|, we get |x| < √5, which means the radius of convergence is √5.
Therefore, the accurate power series representation of f(x) is Σ(-1)^n * x^(2n+1) / 5^(n+1), and the radius of convergence is √5, not 2 as indicated in the question.