Sum: $1 + 3 + 9/2 + 27/6 + 81/24 + \ldots$
This is a problem on a competitive mathematics test, and I am trying to master the concept so I can understand when similar problems show up in future tests. Similar questions may ask for a digit in a different place, or a different type of sum, etc.
Thanks
The Taylor series expansion of $e^x$ about $x=0$ is $$e^x = \sum\limits_{n=0}^{\infty} \frac {x^n} {n!},\ \text {for all }\ x \in \Bbb R.$$
Observe that your sum can be written as $$\sum\limits_{n=0}^{\infty} \frac {3^n} {n!} = e^3.$$