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Program 1

Series Expansion

Set \( n = 50 \) (the upper limit of the natural numbers to sum).

For each \( i \) from 1 to \( n \) (i.e., \( i = 1, 2, 3, \ldots, n \)):

Add \( i \) to total (i.e., total += i).
Append the string representation of \( i \) followed by " + " to series (i.e., series += f"{i} + ").

Print the string series which contains the natural numbers from 1 to \( n \) concatenated with " + " symbols.

Print the value of total, which represents the sum of natural numbers from 1 to \( n \).

Output 1

Program 2

Output 2

Program 3

The exponential function \( e^x \) can be expressed as a Taylor series expansion around \( x = 0 \):

\[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots \]

This series converges for all real values of \( x \) and provides a way to approximate \( e^x \) by summing a finite number of terms.

Variable Initialization:

x = 5: Set the value for which \( e^x \) will be computed.
tol = 1e-16: Define the tolerance level for convergence, ensuring that the algorithm stops when the terms become sufficiently small.
sum, term = 0, 1: Initialize sum to store the cumulative result and term to represent the current term in the series (starting at 1, which corresponds to \( \frac{x^0}{0!} = 1 \)).
n = 1: Initialize the term index, which is used to compute factorials.

The loop continues until the absolute value of the current term is less than the tolerance level. This ensures the approximation is accurate.

Inside the loop:

sum += term: Adds the current term to the cumulative sum.
term = term * x / n: Calculates the next term in the series using the current term. This formula uses the property of the series:

\[ \text{Next term} = \frac{x^n}{n!} = \text{Current term} \times \frac{x}{n} \]

Output 3