BC Calculus: AP Essays on Sequences and Series Worksheet 1

• Class: 5H: BC Calculus
• Author: Peter Atlas
• Text: Calculus Finney, Demana, Waits, Kennedy

Place the solution to each problem on a single, detachable piece of paper, which we can swap around. We will "correct" each others' essays in class.

Questions come from Advanced Placement Calculus tests as indicated.

1988 BC 4: Calculator Inactive

Determine all values of \(x\) for which the series \( \displaystyle \sum_{k = 0}^{\infty} \frac{ 2^kx^k}{\ln {(k + 2)}}\) converges. Justify your answer.

Let \(f\) be the function given by \( \displaystyle f(t) = \frac{4}{1 + t^2}\) and \(G\) be the function given by \( \displaystyle G(x) = \int_0^x f(t) \, dt \).

1. Find the first four nonzero terms and the general term for the power series expansion of \(f(t)\) about \(t = 0\).
2. Find the first four nonzero terms and the general term for the power series expansion of \(G(x)\) about \(x = 0\).
3. Find the interval of convergence of the power series in part (b). (Your solution must include an analysis that justifies your answer.)

Let \(f\) be a function that has derivatives of all orders for all real numbers. Assume \(f(1) = 3, f'(1) = -2, f''(1) = 2\), and \(f'''(1) = 4\).

1. Write the second-degree Tayor polynomial for \(f\) about \(x = 1\), and use it to approximate \(f(0.7)\).
2. Write the third-degree Tayor polynomial for \(f\) about \(x = 1\), and use it to approximate \(f(1.2)\).
3. Write the second-degree Taylor polynomial for \(f'\), the derivative of \(f\), about \(x = 1\), and use it to approximate \(f'(1.2)\).

The Maclaurin series for \(f(x)\) is given by \(1 + \frac{x}{2!} + \frac{x^2}{3!} + \frac{x^3}{4!} + ... + \frac{x^n}{(n + 1)!} + ... \)

1. Find \(f'(0)\) and \(f^{(17)}(0)\).
2. For what values of \(x\) does the given series converge? Show your reasoning.
3. Let \(g(x) = xf(x)\). Write the Maclaurin series for \(g(x)\), showing the first three nonzero terms and the general term.
4. Write \(g(x)\) in terms of a familiar function without using series. Then, write \(f(x)\) in terms of the same familiar function.