expanding exponential of inverse

Question

Expand $(1-3x)^{-1}$ and find the greatest coefficient. Just wondering if there would be need to factorise before proceeding just as in the case of $(2-3x)^{-3}$. Response will be appreciated.

Answer 1

Using Binomial series OR Infinite Geometric Series

$$(1-3x)^{-1}=\sum_{r=0}^\infty(3x)^r$$ for $|3x|<1$

So, the coefficient of $r$th term $T_r=3^r$

$$\implies\dfrac{T_{n+1}}{T_n}=3$$

Clearly, the coefficient are ever increasing.