MATH 181  MIDTERM 4  FALL 1992

Professor Libgober NAME: (PRINT) SIGNATURE: SSN: T.A.: Find the following integrals:

1. For which p, ∑_{n=2}^{∞} \frac{1}{n(ln n)^{p}} converges?

2. For which p, ∑_{n=1}^{∞} n(1 + n^{2})^{p} converges?

MATH 181  MIDTERM 4  FALL 1992

Decide convergence or divergence of the following series and indicate which test you would use.

3. ∑_{n=1}^{∞}(\frac{1 − 3n}{3 + 4n})^{n}

4. ∑_{n=1}^{∞} \frac{1}{\sqrt{n(n + 1)}} .

MATH 181  MIDTERM 4  FALL 1992

5. Decide if the following series converges absolutely, converges conditionally, or diverges.

∑ \frac{(−1)^{n}}{\sqrt[3]{n(n + 1)(n + 2)}}

6. Find the radius of convergence of ∑_{n=1}^{∞} \frac{(−1)^{n}(x − 1)^{n}}{\sqrt{n}} . What is the region of convergence of this series?

MATH 181  MIDTERM 4  FALL 1992

7. Write the first 5 terms of the Taylor series of e^{x} about point c = 3.

8. Find the Maclaurin series of y = cos^{2} x.

MATH 181  MIDTERM 4  FALL 1992

9. Find the Maclaurin series of y = ln(1 + x).

10. Find the first three terms of y = e^{x} ln(1 + x) about c = 0 using multiplication of series.