MAC 2312   Date: 03/29/06  Test 3  period 6

1. Use integration to find the Maclaurin series for \(\ln(1+x)\).
2. Give the sum of the series \(\quad\quad \displaystyle 1+\ln 3+\frac{(\ln 3)^2}{2!}+\frac{(\ln 3)^3}{3!}+\cdots.\)

In 3–6, test each series for convergence or divergence. Use the indicated test.

1. \(\quad\quad\displaystyle \sum_{n=1}^{\infty}\ \frac{n^2+n+2}{(n^2+4) 3^{n}}\quad \)LCT
2. \(\quad\quad\displaystyle \sum_{n=2}^{\infty}\ \frac{1}{(\ln n)^{\ln^n}}\quad \)CT
3. \(\quad\quad\displaystyle \sum_{n=1}^{\infty}\ \frac{\sin \frac{1}{n}}{n} \quad\) LCT
4. \(\quad\quad \displaystyle\sum_{n=1}^{\infty}\ \frac{3^n}{n^n}\quad\)Root Test
5. Find the interval of convergence for \(\displaystyle \sum_{n=2}^{\infty}\ \frac{(x-4)^n}{2^n \ln n}\)
6. \(\displaystyle\sum_{n=1}^{\infty} \sin\frac{1}{n}\quad\) LCT