Calculus I : Midterm Exam Solution April 22, 2014 7:00 PM ∼ 10:00 PM Spring 2014 MAS 101 Full Name(English): Full Name(Korean): Student Number: Class Section: Do not write in this box. Instruction • No items other than scratch papers (given by TA), pen, pencil, eraser and your id are allowed. You must leave all of your belongings that are not allowed at the designated area in the front of the classroom. • Before you start, fill out the identification section on the title page and on the header of page 3 with an inerasable pen. • Show your work for full credit and write the final answer to each problem in the box provided. You are encouraged to write in English but some Korean is acceptable. • The exam is for three hours. Ask permission by raising your hand if you have any question or need to go to toilet. You are not allowed to go to toilet for the first 30 minutes and the last 30 minutes. • Any attempt to cheat or a failure to follow this instruction lead to serious disciplinary actions not limited to failing the exam or the course. Do not write in this table. Problem Score Problem Score Problem Score 1 / 48 5(a) /4 6(b) / 10 2 / 20 5(b) /4 7 / 20 3 / 20 5(c) /4 8 / 20 4 / 20 6(a) / 10 9 / 20 Total Score: –1–
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Calculus I : Midterm Exam Solutionwingdev.kr/zokbo/file/KAIST/Common/MAS101/Midterm... · 2014 Calculus 1 Mid-term Solution TA Jonghun Yoon 1 48 points (a) If P (a n+b n) converges
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• No items other than scratch papers (given by TA), pen, pencil, eraser and your id areallowed. You must leave all of your belongings that are not allowed at the designatedarea in the front of the classroom.
• Before you start, fill out the identification section on the title page and on the header ofpage 3 with an inerasable pen.
• Show your work for full credit and write the final answer to each problem in the boxprovided. You are encouraged to write in English but some Korean is acceptable.
• The exam is for three hours. Ask permission by raising your hand if you have any questionor need to go to toilet. You are not allowed to go to toilet for the first 30 minutes andthe last 30 minutes.
• Any attempt to cheat or a failure to follow this instruction lead to serious disciplinaryactions not limited to failing the exam or the course.
Do not write in this table.Problem Score Problem Score Problem Score
1 / 48 5(a) / 4 6(b) / 10
2 / 20 5(b) / 4 7 / 20
3 / 20 5(c) / 4 8 / 20
4 / 20 6(a) / 10 9 / 20
Total Score:
– 1 –
2014 Calculus 1 Mid-term Solution TA Jonghun Yoon
1
48 points
(a) If∑
(an + bn) converges and∑
an diverges, then∑
bn diverges.
(b) If all an are positive and an+1
an< 1 for all n, then
∑an converges.
(c) If all an are positive and n√an < 1 for all n, then
∑an converges.
(d) If 0 < an+1 < an for all n, then∑
(−1)nan converges.
(e) If∑
an converges, then∑|an| converges.
(f) If∑
an converges, then∑
(−1)nan converges.
(g) If∑
an converges, then∑
a2n converges.
(h) If the an are positive and∑
an converges, then∑
a2n converges.
(i) (x+ sinx) · coshx = o (xx) as x→∞.
(j) 2(x+ (lnx)2) tan−1 (ex) = O (x− lnx) as x→∞.
(k)∫∞1
ln xx2 dx converges.
(l) The domain of coth−1 x is (−∞,−1) ∪ (1,∞)
Solution.
(a) T, If∑
bn converges, then∑
an =∑{(an + bn)− bn} converges.
(b) F, The counter example is an = 1/n.
(c) F, The counter example is an = 1/(n+ 1).
(d) F, The counter example is an = 1 + 1/n.
(e) F, The counter example is an = (−1)n/n.(f) F, The counter example is an = (−1)n/n.(g) F, The counter example is an = (−1)n/
√n.
(h) T, Since∑
an converges, limn→∞a2nan
= 0. Hence, for all large n, 0 < a2n ≤ Man for some
M > 0.
(i) T, Use limx→∞( ex )x = 0. This is true since e
x < 12 for all large x.
(j) T, By L’ohospital’s rule, limx→∞x+(ln x)2
x−ln x = 1. And | tan−1 x| ≤ π2 .
(k) T, By Integral by parts,∫ a1
ln xx2 dx =
[− ln x
x −1x
]a1= − ln a
a −1a + 1. Hence
∫∞1
ln xx2 dx = 1
(l) T, See the text book 439 page.
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2014 Calculus 1 Mid-term Solution TA Jongbaek Song
2
20 points
Decide whether the infinite series
∞∑n=1
(−1)n(√
n5 + n2 −√n5),
converges absolutely, converges conditionally, or diverges. You should justify your
answer.
Solution. Let an =√n5 + n2 −
√n5 = n2
√n5+n2+n5
= 1√n+ 1
n2 +√n.
• [9 points]∑∞
n=1 an diverges.
Indeed, use the sequence bn = 1√nto apply the limit comparison test,
limn→∞
anbn
=1
2.
Since∑∞
i=11√ndiverges by p-series, so does
∑∞n=1 an by limit comparison test.
• [9 points]∑∞
n=1(−1)nan converges. Use the following alternating series test (Leibniz’s test).
1. an ≥ 0, for all n, obviously. · · · · · · (3 points)
2. an = 1√n+ 1
n2 +√n≥ 1√
n+1+√n+1
≥ 1√n+1+ 1
(n+1)2+√n+1
= an+1
Hence, an is a non-increasing sequence. · · · · · · (3 points)
3. an = 1√n+ 1
n2 +√n→ 0 as n → ∞. · · · · · · (3 points)
• [2 points] Therefore,∑∞
n=1(−1)nan converges conditionally.
Grading Policy
• You may use other methods to show that∑∞
n=1 an diverges or an is an decreasing sequence.
• If you just state something without any explanation, you wouldn’t get the associated points.
Especially, you have to explain why the sequence an is decreasing.
• 2 points assigned in the final conclusion would follow only when your solution explain that.
In other words, If your solution has nothing to do with conditionally convergence, then 2
points might not be given, even though you conclude correctly.
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Calculus I Midterm Solution and Scoring Criteria TA. Giwon Suh