Taylor Series Problem

Question

I do not understand how to solve this.

Answer 1

Hi Shannen! Just apply the definition of a Taylor series at x = 7.

f(x) = x^0.8, so c₀ = f(7)/0! =7^0.8

f'(x) = 0.8x^{0.8 - 1}, so c₁ = f'(7)/1! = 0.8*7^{0.8 - 1}/1!

f''(x) = 0.8(0.8 - 1)x^{0.8 - 2}, so c₂ = f''(7)/2! = 0.8(0.8 - 1)7^{0.8 - 2}

Continuing in this fashion, we find that in general:

f⁽ⁿ⁾(x) = 0.8(0.8 - 1)*...*(0.8 - n + 1)x^{0.8 - n}, so c_n = 0.8(0.8 - 1)*...*(0.8 - n + 1)7^{0.8 - n}

So the power series is as follows.

f(x) = ∑0.8(0.8 - 1)*...*(0.8 - n + 1)7^{0.8 - n}(x - 7)ⁿ

The sum goes from 0 to infinity.