Discuss the difference between truncation and rounding errors, and provide an example of each.

Truncation Error:

• Definition: This error results from stopping an infinite mathematical process (like an infinite series or an integral) after a finite number of steps, or from approximating a continuous function with a discrete one. It's an error due to the method or model used.
• Example: Approximating the value of $e^x$ using only the first few terms of its Taylor series expansion. The Taylor series for $e^x$ is $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$ If we approximate $e^1$ using only the first three terms: $e^1 \approx 1 + 1 + \frac{1^2}{2!} = 1 + 1 + 0.5 = 2.5$ The true value of $e^1$ is approximately 2.71828. The truncation error is $2.71828 - 2.5 = 0.21828$. This error arises because we "truncated" the infinite series.

Rounding Error:

• Definition: This error occurs when a number with more significant digits is represented by a number with fewer significant digits, typically due to limitations in storage or display. It's an error due to the representation of a number.
• Example: Representing the fraction $\frac{1}{3}$ as a decimal with a limited number of digits. The true value of $\frac{1}{3}$ is $0.333333\dots$ (an infinitely repeating decimal). If we round $\frac{1}{3}$ to two decimal places, we get 0.33. The rounding error is $\frac{1}{3} - 0.33 = 0.333333\dots - 0.33 = 0.003333\dots$