Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due

Math 1152 - Written Homework 2 - Spring 2025

1) Directions: Fill in the circle next to the correct response(s).

a) (6 points) Multiselect Fill in the circle next to each improper integral below. There may be multiple answers.

b) (10 points) True or False Determine whether each statement below is true or false.

2) (9 points) For each sequence below, determine the limit as n → ∞. If a limit is infinite, write “∞” or “−∞” as appropriate.

You do not need to show your work, and there is no partial credit. Write only your final answer in the box provided.

3) (6 points) State the correct general form of the partial fraction decomposition for

Do NOT solve for the constants.

4) (6 points) State the limit or sum of limits of proper integrals that must be evaluated in order to determine whether the improper integral  dx converges or diverges. Do not evaluate any integrals or compute any limits that you write down.

5) (6 points) Let  Determine  Explicitly indicate dominant terms when taking the limit.

6) (16 points) Consider the function 

a) (6 points) Determine the partial fraction decomposition of  You must show all the intermediate steps.

b) (10 points) Consider the improper integral 

Determine if this integral converges or diverges. If it converges, state its value. In order to earn full credit, you must show all your work, justify any limits you compute and use proper notation!

7) Complete the following parts to explore and better understand

❼ The sequence bn

❼ The sequence  of partial sums

❼  the limit of the sequence of partial sums

a) (6 points)  Use partial fractions to show that  Show all intermediate steps.

b) (3 points) Write out the first 6 terms of  Do not simplify.

b1 = b4 =

b2 = b5 =

b3 = b6 =

c) (2 points) Modify the Desmos Interactive found at https://www.desmos.com/calculator/ljnyljth5b to graph the se quence bn. Based on your graph, estimate .

 =

d) (2 points) Determine  directly using the formula for bn. Write a sentence to explain your reasoning.

e) (4 points) Let  Use the definition of s6 and your work in part (b) to write out an unsimplified expression for s6. Then show how to simplify this sum to obtain 

f) (6 points) Look for the pattern in your work in part (e) and then use the pattern to write an explicit formula for sn. Make sure you show how you obtain this explicit formula from the definition of sn. Then determine a function f(x) of a continuous variable x such that f(n) = sn when n is a natural number.

The explicit formula of sn =                                  f(x) =

g) (4 points) Modify the Desmos Interactive found at https://www.desmos.com/calculator/q6pmypnurl to:

❼ Graph the sequence bn

❼ Graph the sequence of partial sums sn

❼ Graph the function f(x) you found in part (f).

❼ Graph the horizontal asymptote of f as x → ∞.

Paste a screenshot of your graph from Desmos below.

h) (2 points) Use your graph in Desmos to estimate  =

i) (4 points) Determine  by using the formula you found in part (f). Write a sentence or two explaining your reasoning.