If ∑ n=0 to inf of 3x^n/n! is a Taylor series that converges to f(x) for all real x, what is the value of f ''(0)?
2 answers
0^anything = 0
expanding the series, we have
f(x) = 3/0! x^0 + 3/1! x^1 + 3/2! x^2 + 3/3! x^3 + ...
f'(x) = 0 + 3/1! + 3*2/2! x + 3*3/3! x^2 + ...
f"(x) = 0 + 0 + 3*2/2! + 3*6/3! x^1 + ...
So, f"(0) = 3
f(x) = 3/0! x^0 + 3/1! x^1 + 3/2! x^2 + 3/3! x^3 + ...
f'(x) = 0 + 3/1! + 3*2/2! x + 3*3/3! x^2 + ...
f"(x) = 0 + 0 + 3*2/2! + 3*6/3! x^1 + ...
So, f"(0) = 3