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Anonymous

hello

calculus 2 help needed

show work like a((( Student solving ))))NOT like math professional SOLVING

topics of questions : Series , the integral test , sequences

total 7 question in the attachment bellow

Math 156
Spring 2020
Test 3 – Part A
This test has a total of 43 points, but is only worth 40 points. It is possible to get 3
points of extra credit.
• Follow instructions for each problem carefully.
• Show all of the steps. No credit will be given for answers without supporting work.
• You must submit your own work.
• Make sure that your solutions are neat and your files are readable (not faint or blurry) and
oriented correctly (not sideways).
(
1. (7 points) For the sequence
(−1)n+1 (2n − 1)
n2 + 2
)∞
, do all of the following.
n=1
(a) (2 points) Write the first four terms of this sequence.
(b) (3 points) Determine whether this sequence is convergent or divergent. If it is convergent, find
the limit. If it is divergent, justify your answer.
(c) (2 points) Determine whether this sequence is increasing, decreasing, or not monotonic. Justify
your answer.
2. (6 points) Find the limit of the sequence an =
ln(5n + 3)
e2n−4
Show all of your work and use the correct notation.
3. (7 points) If the n-th partial sum of a series
∞
X
an is sn =
n=1
2n + 1
, find the following.
7n − 2
(You do not have to simplify your answers, but it must be clear how you obtained them.)
(a) a1 =
(b) a2 + a3 + a4 + a5 =
(c)
∞
X
an =
n=1
(d) Is the series
∞
X
sn convergent or divergent. Justify your answer.
n=1
4. (8 points) For the series
∞
X
n2
n3 + 2
n=3
(a) Verify that the integral test can be used to test the convergence of this series. Check all
necessary conditions. Show all work.
(b) Perform the integral test to determine convergence of this series. Clearly state your conclusion.
3n − 1 ∞
5. (6 points) Determine whether the sequence
is increasing, decreasing, or not monon + 1 n=1
tonic. You must show work to support your conclusion.
6. (6 points) Find the sum of the following series. Show all algebraic steps. Simplify your answer
completely.
∞
X
3 + (−4)n
n=1
5n
7. (3 points) Determine whether the following statement is true or false.
If this statement is always true, write TRUE and provide an explanation.
If the statement is false, write FALSE and provide a counterexample.
Note: a counterexample is a specific example that shows that the statement is false. Your
counterexample must be simple and obvious.
If a sequence {an } is bounded, then it is convergent.
...

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