Taylor and Maclaurin series

Question

Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that R_n(x) → 0.]

f(x) = 8^x

f(x) = ∑ (from n=0 to ∞) ( ? )

Also find the associated radius of convergence R.

R = ?

Answer 1

The Maclaurin Series is f(0) + f'(0)x + f"(0)/2! x² + f^'''(0)/3!x³ + ...

f(x) = 8^x, so f(0) = 1

f'(x) = (ln8)8^x, so f'(0) = ln8

f"(x) = (ln8)(ln8)8^x, so f"(0) = (ln8)², etc

So, we have 1 + (ln8)x + (ln8)²x² / 2! + (ln8)³x³ / 3! + ...= ∑_{(n=0 to ∞)} [(xln8)ⁿ / n!] (Recall 0! = 1! = 1)

Use Ratio Test to get interval of convergence:

l [(xln8)ⁿ⁺¹/(n+1)!][n!/(xln8)ⁿ] l = (1/(n+1))ln8 lxl

lim_n→∞ [(1/(n+1))ln8 lxl] = lxl (ln8)(0) = 0 < 1 for all real x.

So, interval of convergence = (-∞, ∞) and R = radius of convergence = ∞.