Series Assignment.sagews

Series Assignment

Question 0

Watch the lecture video here.

Did you watch the video? [Type yes or no.]

Question 1

Find the sum of the following geometric series in two ways:

1. Using the sum command in Sage.

2. Using the formula $\displaystyle\frac{a}{1-r}$.

(You should get the same answer).

Part a

$\displaystyle 1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots$

Part b

$\displaystyle \frac{1}{2}+\frac{1}{8}+\frac{1}{32}+\frac{1}{128}+\cdots$

Part c

$\displaystyle \sum_{n=0}^{\infty} 4\left(-\frac{1}{5}\right)^n$

Question 2

Consider the series $\displaystyle \sum_{n=1}^{\infty}\left(n\cdot r^n\right)$.

Part a

Find the sum of this series for the following values of $r$: $\displaystyle -2,\ -1,\ -\frac{7}{8},\ -\frac{1}{2},\ \frac{1}{2},\ \frac{7}{8},\ 1,\ 2$.

Part b

Make a conjecture (an educated guess): for what values of $r$ does this series converge?

[Note: I'm asking about $r$ in general, not the values from part a.]

Question 3

Check that $\displaystyle\sum_{n=1}^{\infty}\frac{\sin(n)}{n}=\frac{\pi-1}{2}$.

(Sage will give you some strange-looking output, but you can simplify it, since $\tan=\sin/\cos$.)

Question 4

Find the 10th, 100th, and 1000th partial sums of the series $\displaystyle\sum_{n=1}^{\infty}\frac{n!}{n^n}$. Do you think the series converges?

[Use factorial(n) for $n!$]

[Hint: You'll want to convert to a decimal using the N() command.]

Question 5

Find the 10th, 100th, and 1000th partial sums of the series $\displaystyle\sum_{n=1}^{\infty}\frac{n^n}{n!}$. Do you think the series converges?

[Hint: Watch out for scientific notation in the answers.]

This material was developed by Aaron Tresham at the University of Hawaii at Hilo and is
licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.