{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.
∫₀⁰ᐧ² (ln (1 + t))/t dt

Verified step by step guidance

1. Recognize that the integral involves the function \( \frac{\ln(1+t)}{t} \). To approximate this integral using a Taylor series, start by finding the Taylor series expansion of \( \ln(1+t) \) around \( t=0 \).
2. Recall the Taylor series for \( \ln(1+t) \) is \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{t^n}{n} \) for \( |t| < 1 \). Write this series explicitly: \( \ln(1+t) = t - \frac{t^2}{2} + \frac{t^3}{3} - \frac{t^4}{4} + \cdots \).
3. Divide the series for \( \ln(1+t) \) by \( t \) to get the series for \( \frac{\ln(1+t)}{t} \): \( \frac{\ln(1+t)}{t} = 1 - \frac{t}{2} + \frac{t^2}{3} - \frac{t^3}{4} + \cdots \).
4. Integrate the series term-by-term from 0 to 0.2: \[ \int_0^{0.2} \frac{\ln(1+t)}{t} dt = \int_0^{0.2} \left(1 - \frac{t}{2} + \frac{t^2}{3} - \frac{t^3}{4} + \cdots \right) dt = \sum_{n=0}^{\infty} a_n \int_0^{0.2} t^n dt, \] where \( a_n \) are the coefficients from the series.
5. Calculate the integral of each term: \( \int_0^{0.2} t^n dt = \frac{(0.2)^{n+1}}{n+1} \). Sum the terms until the absolute value of the next term is less than \( 10^{-4} \) to ensure the error is within the required tolerance.

Key Concepts

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. It allows approximation of complex functions by polynomials, which are easier to integrate or evaluate. For ln(1 + t), the series expansion around t = 0 is especially useful for approximating integrals.

Definite Integral Approximation Using Series

When a function is expressed as a Taylor series, its definite integral over an interval can be approximated by integrating the polynomial term-by-term. This method simplifies the integral calculation and provides a way to estimate the value with controllable accuracy by including enough terms.

Error Estimation and Convergence Criteria

To ensure the approximation is accurate within a specified error bound (e.g., less than 10⁻⁴), it is essential to estimate the remainder or error term of the Taylor series. Understanding how to bound this error helps determine how many terms to retain for a reliable approximation.