Find a power series representation for the function; find the interval of convergence. (Give your power series representation centered at x = 0.)

Question

Find a power series representation for the function; find the interval of convergence. (Give your power series representation centered at x = 0.)

f(x)= ln(1-8x)

sum from n=0 to infinity [?] provided |x| < [?]

Answer 1

Let take function 8/(1-8x) we can considered as sum of infinite geometric series with a₁ = 8 and r = 8x.

Then 8/(1 - 8x) = 8 + 8²x + 8³x^2 + ... + 8^(n + 1)x^n + ... = ∑₀^∞8^{n + 1}xⁿ; Now we integrate both sides from 0 to x; ∫₀^x8/(1 - 8x)dx = ∑₀^∞∫₀^x8ⁿ⁺¹⁾xⁿdx; or - ln(1 - 8x) = ∑₀^∞8^{n + 1}x^{n + 1}/(n + 1); from here

f(x) = ln(1 - 8x) = - ∑₀^∞8^{n + 1}x^{n + 1}/(n + 1);

Interval of convergence we get from condition - 1 < r < 1: - 1< 8x < 1; from here - 1/8 < x < 1/8 ,

x ∈ (-1/8, 1/8)