Assignment: Calculating $e^x$ using Convergent Series

Objective:

Understand the concept of convergent series.

Implement a Python program to calculate $e^x$ using the series expansion.

Analyze the behavior of the series for $x=-25$ with different values of $n.$

Instructions:

Part $1$: Convergent Series

The series expansion for $e^x$ is given by:

$e^z=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+$dots

Here, $n!$ is the factorial of $n,$ defined as $nn*(n-1)*(n-2)*dots*2*1.$

Part $2$: Python Program

Write a Python program to compute the partial sum of the series for $e^{-25}$ using various

values of $n.$ Use the formula:

$S_n=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+$dots$+\frac{x^n}{n!}$

Your program should:

Accept the value of $x$ and $n$ as inputs.

Compute the partial sum $S_n$ for the given $x$ and $n.$

Display the result.

Part $3$: Analysis

After running your program, observe the results for different values of $n$ when $x=-25.$

Answer the following questions:

How does the value of $S_n$ change as $n$ increases?

Is the result ever close to $e^{-25}?$ Explain your observation.

Submission Guidelines:

Submit your Python program code.

Include a brief report addressing the analysis questions.

This assignment aims to reinforce the understanding of convergent series and provide

students with hands$-$on experience in implementing a numerical computation using

Python. It also encourages critical thinking about the behavior of the series for a

specific value of $x.$