Convergent and divergent infinite series; geometric series：练习题

Determine whether the series \(\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}\) converges or diverges, and identify the test used to justify the conclusion.

The Taylor series for a function \(f\) centered at \(x=0\) is given by \(f(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n}\). Which of the following is the value of \(f^{(4)}(0)\)?

Find the third-degree Taylor polynomial for \(f(x) = \ln(x)\) centered at \(x = 1\).

Which of the following represents the sum of the convergent geometric series \(\sum_{n=0}^{\infty} 3\left(-\frac{2}{5}\right)^n\)?

Find the interval of convergence for the power series \(\sum_{n=1}^{\infty} \frac{(x-2)^n}{n \cdot 3^n}\).

Determine the interval of convergence of the power series given by \(\sum_{n=1}^{\infty} \frac{(-1)^{n}(x-3)^{n}}{n \cdot 4^{n}}\). Be sure to show the testing of the endpoints and state the final interval in interval notation.

Consider the function defined by \( f(x) = \frac{x}{1 + 2x^3} \).

(a) Write the first four non-zero terms and the general term for the Maclaurin series of \( f(x) \). Determine the interval of convergence for this series, showing the work that leads to your conclusion.
(b) Let \( g \) be the function defined by \( g(x) = \int_0^x f(t) dt \). Find the first four non-zero terms of the Maclaurin series for \( g(x) \).
(c) Use the first three non-zero terms of the Maclaurin series found in part (b) to approximate the value of \( g\left(\frac{1}{2}\right) \).
(d) Use the Alternating Series Error Bound to determine if the absolute error of the approximation found in part (c) is less than \( 5 \times 10^{-4} \). Justify your answer.