write the first 4 terms of the taylor's series for f(x)=3^x centered at c=0
recall that the Taylor series is of the form:
$\displaystyle \sum_{n=0}^{ \infty} \frac {f^{(n)}(x_0)}{n!} (x - x_0)^n$ where the series is centered at $\displaystyle x_0$
the first four terms of the Taylor series is centered at 0 is given by:
$\displaystyle \sum_{n=0}^{3} \frac {f^{(n)}(0)}{n!} x^n$
$\displaystyle f(x) = 3^x \Rightarrow f(0) = 1$
$\displaystyle f'(x) = ln(3) \cdot 3^x \Rightarrow f'(0) = ln(3)$
$\displaystyle f''(x) = (ln(3))^2 \cdot 3^x \Rightarrow f''(0) = (ln(3))^2$
$\displaystyle f'''(x) = (ln(3))^3 \cdot 3^x \Rightarrow f'''(0) = (ln(3))^3$
So the first 4 terms of the Taylor series is:
$\displaystyle 1 + ln(3) x + \frac {(ln(3))^2}{2!}x^2 + \frac {(ln(3))^3}{3!}x^3 $