Sections 5 and 6

The part on convergence of series is a bit small for my taste. There are so many different possible examples for each convergence test, but you have chosen just one or two. I will be more imaginative in my final. Please do work some extra problems on this topic from the assigned HW in the syllabus, and online.

Same for sequences. I wish I had seen some more interesting examples, with varied difficulty and challenges.

The problems for power series that you presented are really good. There is a good variety in the selection, with different difficulty levels. There is one evil problem out there (thanks Emmy?). Unfortunately, you have missed about 50% of the material. I would have liked to see more problems on finding the interval of convergence of a few series that are not expressed as standard functions. I did modify one of the problems to address this situation. As before, I will be much more imaginative in my final.

1. Explore the convergence (absolute or conditional) or divergence of the following series.
   \begin{align} & \sum_{n=0}^\infty \frac{6^n+8}{11^{n+1}+27} & & \sum_{n=0}^\infty \frac{4n^4+4n^2+3}{\sqrt{n^7+5}} & & \sum_{n=0}^\infty \frac{(-1)^n}{5n+2} \\ & \sum_{n=0}^\infty (-1)^n \frac{n}{\sqrt{n^3+8}} & & \sum_{n=1}^\infty \bigg( \frac{6n+2}{3n+5} \bigg)^n & & \sum_{n=1}^\infty (-1)^n \frac{3n^2+4}{n!} \\ & \sum_{n=1}^\infty \frac{\ln n}{n} & & \sum_{n=50}^\infty ne^{-n} \end{align}
2. Find the following sums (an exact value). Show work.
   \begin{align} \sum_{n=1}^\infty \frac{(2e)^n}{6^{n-1}} && \sum_{n=1}^\infty \frac{1}{(n+2)(n+3)} \end{align}

3. Find a formula for the general term of the sequence that starts \( 4, -\frac{8}{5}, \frac{16}{25}, -\frac{32}{125}, \dotsc \)

4. Study the convergence of the following sequence.
   \begin{equation} \left\{ \frac{5e^n + e^{-2n}}{6+e^{3n}} \right\}_{n=1}^\infty \end{equation}

5. Express the following sums as power series, and indicate what is the center and radius of convergence.

   \begin{align} f(x) &= \frac{1}{5-x} & f(x) &= \frac{1}{(1-x)^2} \\ f(x) &= 7\pi \tan^{-1} x & f(x) &= \frac{x^2}{(1-x)^3} \\ f(x) &= \frac{3}{x^2-x-2} & f(x) &= \frac{x^8}{x^2-4} \end{align}

6. Find the interval of convergence of the following series:

   \begin{equation} \sum_{n=0}^\infty \frac{n^n x^n}{n!} \end{equation}
7. Find a Taylor series representation of \( f(x) = \sin x \) around \( a=\pi/2 \).