Question #1:
Estimate the following sums to at least 6 decimal places:
$$\sum_{i \geq 1} 1/i^{4/3} = 1 + 2^{-4/3} + 3^{-4/3} + 4^{-4/3} + \cdots$$
$$\sum_{i \geq 1} (-1)^{i+1}/i^{4/3} = 1 - 2^{-4/3} + 3^{-4/3} - 4^{-4/3} + \cdots$$
$$\sum_{i \geq1} (3i-2)^{-5/2} = 1 + 1/4^{5/2} + 1/7^{5/2} + \cdots$$
Find a formula in terms of $n$ for the following sums:
$$\sum_{i =1}^{n} i^5 = 1^5 + 2^5 + 3^5 + \cdots + n^5$$
$$\sum_{i =1}^{n} (3i-2)^5 = 1^5 + 4^5 + 7^5 + \cdots + (3n-2)^5$$
The sum $\sum_{i\geq1}1/i^{5/6}$ diverges. Find the minimum number of terms that you need to
add together before the sum is greater than $12.5$. Find the minimum number of terms that you need to
add together before the sum is $1000$. You may find that it is difficult to solve this
problem precisely. In this case tell me what you tried and give me the best range
that you can for the minimum number of terms for adding up to 1000.
Question #2:
Plot the two polar equations $r=2+sin(8\theta)/2$ and $r=3+cos(\theta)$. You can graph them with
the following two commands in Maple (one will be in blue, the other in red).
Now display these plots on the same graph and find the area of the region shaded by
blue, red and purple, each to at least 6 decimal places.
Maple 1-15 has a bug in it. Maple 16 seems to plot these graphs correctly. What you should
be seeing is pictured below.
In order to determine the area of these regions you are going
to have to dust off a little calculus which says that the area of the sector traced
out by a curve $r(\theta)$ from $\theta_1$ to $\theta_2$ is equal to
\[ \int_{\theta_1}^{\theta_2} \frac{1}{2} r(\theta)^2 d\theta ~. \]
You will also need to numerically solve for the intersection points of these curves.
Explain what you are doing in your calculations. Identify your intersection points.
Label your answers and tell me your findings. How do you know you are correct?
You should open up a new worksheet and start from scratch. You will have to save
your work in a file and upload that file on to the course
moodle. Your
solution should be a sequence of commands where it is easy to change the input
string and after you execute the sequence of commands you should have the
correct output string. Please add documentation to your worksheet to explain how it
works. Just a few sentences is sufficient, but imagine that you were opening up the
worksheet for the first time and wanted to know what it did. You will be marked down
if what you write is not clear and coherent.
You should finish your assignment by Wednesday, November 14 by 11:59pm. Assignments
submitted after this date will be assessed a penalty of 10% per day.
