Question

Let f(x) be the function defined by the power series sum n=0 to infinity of 2x^n. If g'(x) = f(x) and
g(0) = 2, then find g(x).

Answer 1

\[f(x) = \sum_{n=0}^{\infty} 2x ^{n} \]

Answer 2

That is easy. You should be able to do it.

Answer 3

\[
g(x) =2 -2\ln(1-x),\quad |x|<1

\]

Answer 4

\[
g(t)=\int_0^t \left(\sum _{n=0}^{\infty } 2
x^n\right) \, dx= \left(\sum
_{n=0}^{\infty } \int_0^t 2 x^n \,
dx\right)=\sum _{n=0}^{\infty } \frac{2
t^{n+1}}{n+1}+C=C-2 \ln (1-t)
\]
Since g(0)=2, youget C=2

Answer 5

thanks!