Calculus II, Fall 2013

Quiz 2

Mar 21, 2013 Show all details.

1. For what values of x does the series ∑_{n=2}^{∞} \frac{x^{n}}{n ln n} converge? Explain.

2. Suppose that f is infinitely many times differentiable on R. State Taylor’s Theorem for f and derive the formula for R_{2}(x) (in any form you choose).

3. Find the Taylor series generated by sin x at x = 0. For what values of x does sin x equal this series? Explain.

4. Find the sum of the series

\frac{1}{2} − \frac{1}{8} + \frac{1}{24} − ⋯ + (−1)^{n−1} \frac{1}{n2^{n}} + ⋯

5. Let f(x) = exp(\frac{−1}{x^{4}}) for x ≠ 0 and f(0) = 0. Is f differentiable everywhere? Explain.

Calculus II, Fall 2013

Quiz 2

Mar 21, 2013 Show all details.

1. For what values of x does the series ∑_{n=2}^{∞} \frac{x^{n}}{n ln n} converge? Explain.

2. Suppose that f is infinitely many times differentiable on R. State Taylor’s Theorem for f and derive the formula for R_{2}(x) (in any form you choose).

3. Find the Taylor series generated by sin x at x = 0. For what values of x does sin x equal this series? Explain.

4. Find the sum of the series

\frac{1}{2} − \frac{1}{8} + \frac{1}{24} − ⋯ + (−1)^{n−1} \frac{1}{n2^{n}} + ⋯

5. Let f(x) = exp(\frac{−1}{x^{4}}) for x ≠ 0 and f(0) = 0. Is f differentiable everywhere? Explain.