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## Homework Statement

"Aim: In this task, you will investigate the sum of infinite sequences t

_{n}, where

t

_{n}= [tex]{\frac{(x\ln{a})^n}{n!}}[/tex], and t

_{0}=1

Consider the sequence when x=1 and a=2.

Using technology, plot the relation between S

_{n}(the sum of t

_{0}+...+t

_{n}) and the first

*n*terms of the sequence for [tex]{0}\leq{n}\leq{10}[/tex].

What does this suggest about S

_{n}as

*n*approaches [tex]\infty[/tex]?"

## The Attempt at a Solution

We were given this assignment (which originally is much lengthier than this, but this is the core idea of it) with the prospect of using a calculator and/or graphing software to deduce that the value of the sum of this sequence converges as n becomes arbitrarily large. We have never been exposed to infinite sums (or even calculating the value of non-infinite sums) and are expected to do this completely dry, not using any formal analysis of the sum. I'm suspicious that this sum,

[tex]\sum\limits_{n=0}^{\infty}{\frac{(x\ln{a})^n}{n!}}[/tex]

has an analytic solution, and I'd love to find out what it is. I'm also taking AP Calculus BC as a parallel to this class, but we haven't come close to infinite sums yet. I've done a lot of integration and significantly more differential calculus independently. I was wondering, what direction should I take if I wanted to include an analytical solution for this convergence (not just plugging in values into a calculator and seeing where it leads, as my school wants me to)? Thanks, any help appreciated.