# 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.

1991 BC 5: Calculator Inactive

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.)

1995 BC 4: Calculator Inactive

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 $$f'(1.2)$$.

1996 BC 2: Calculator active

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.