Let f(x) = the summation from n equals 0 to infinity of a sub n times x to the nth power and g '(x) = f(x). Find g(x) if g(0) = 4

Question

Let f(x) =  and g '(x) = f(x). Find g(x) if g(0) = 4

A
B
C
D

Answer 1

To start, we note that the integral of xⁿ is xⁿ⁺¹/(n+1)+C. Thus, since integrals are linear, that is the integral of a sum is the sum of the integrals, we see that the indefinite integral of f(x) is the sum  from 0 to infinity of a_n/(n+1) * x⁽ⁿ⁺¹⁾+C. Then, if we reindex in terms of k=n+1, then it is the sum from k=1 to infinity of a_k-1/k * x^k+C. Finally, but plugging in zero for x, we see that C=4, so the answer is D.