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Determine whether the following series converge or diverge. Practice writing out your work carefully for each problem (though you don't have to submit it). State which test you are using, show all your computations, and state your final answer (converges/diverges).

1. \(\sum_{n = 1}^{\infty} n (\frac{2}{3})^n\)

a.  The series converges.

b.  The series diverges.

2. \(\sum_{n = 1}^{\infty} \frac{1}{n+\sqrt{n}}\)

a.  The series converges.

b.  The series diverges.

3. \(\sum_{n = 1}^{\infty} \frac{(n!)^2}{(3n)!}\)

a.  The series converges.

b.  The series diverges.

4. \(\sum_{n = 1}^{\infty} \frac{1}{3n}\)

a.  The series converges.

b.  The series diverges.

5. Can the series \(\sum_{n = 1}^{\infty} (-1)^n\frac{1}{n^2}\) be rearranged to get any limit you want?

a.  Yes, because it converges absolutely.

b.  No, because it converges absolutely.

a.  Yes, because it converges conditionally.

b.  No, because it converges conditionally.

6. Can the series \(\sum_{n = 1}^{\infty} (-1)^n\frac{1}{\sqrt{n}}\) be rearranged to get any limit you want?

a.  Yes, because it converges absolutely.

b.  No, because it converges absolutely.

a.  Yes, because it converges conditionally.

b.  No, because it converges conditionally.

7. Find the taylor polynomial of order 3 around \(x = 0\) for the function \(f(x) = e^{2x}\).

a.  \(\frac{1}{3}x^3 +\frac{1}{2}x^2 + x + 1\)

b.  \(\frac{4}{3}x^3 + 2x^2 + 2x + 1\)

a.  \(\frac{1}{6}x^3 +\frac{1}{2}x^2 + x + 1\)

b.  \(\frac{1}{6}x^3 +\frac{1}{2}x^2 + 2x + 1\)

a.  \(\frac{4}{3}x^3 + 2x^2 + x + 1\)