91Ó°ÊÓ

Series expansion is a powerful tool in mathematics that allows us to express complex functions as infinite sums of simpler terms. Specifically, for this exercise, we use the series expansion for \( \ln(1+t) \), which is given by:

• \( \ln(1+t) = t - \frac{t^2}{2} + \frac{t^3}{3} - \frac{t^4}{4} + \ldots \)

This expansion is valid for \(-1 < t \leq 1\), which suits our problem where the integration interval starts at 0. The beauty of series expansion lies in its flexibility. By truncating the series after a few terms, we obtain a polynomial approximation of the actual function.

Substituting this series into the integral's integrand simplifies the function, allowing us to perform integration term-by-term. This is particularly useful in cases where the integral does not have a simple closed-form solution. You'll see that each term in the expansion corresponds to a simple polynomial expression, making it easier to work with.

Integral calculus is the branch of calculus that deals with integrals and their properties. In our exercise, we are evaluating the integral \( F(x) = \int_{0}^{x} \frac{\ln(1+t)}{t} \, dt \). This integral represents the accumulation of the area under the curve described by our function.

When we substitute the series expansion into the integral, it transforms the original complex problem into multiple simpler problems. Each term in the series leads to a corresponding integral:

• \( \int_{0}^{x} 1 \, dt = x \)
• \( \int_{0}^{x} -\frac{t}{2} \, dt = -\frac{x^2}{4} \)
• \( \int_{0}^{x} \frac{t^2}{3} \, dt = \frac{x^3}{9} \)
• and so on.

This method is known as term-by-term integration, and it simplifies the process significantly. Once these individual integrals are calculated, they can be summed to form a polynomial that approximates the integral function \( F(x) \). The resulting polynomial provides an estimate of the area under the original curve to a specified accuracy.

Error estimation is crucial when approximating functions using polynomials. The main goal is to ensure that the polynomial approximation is within a specific range of accuracy—in our case, less than \(10^{-3}\).

To achieve this, it's important to determine where we can truncate the series without significantly affecting accuracy. This involves calculating the magnitude of the next term in the series that we decided to omit. For smaller intervals like \([0, 0.5]\), fewer terms might be necessary than for larger intervals such as \([0, 1]\). Typically, you continue adding terms until the omitted term becomes less than \( 10^{-3} \) over the interval of interest.

• Calculate the magnitude of the next omitted term.
• Compare it to the desired error bound.
• Continue including terms in the polynomial until the condition is satisfied.

This approach guarantees that the resulting polynomial is an accurate approximation of the function within the specified error limit, ensuring precision over the entire interval.