CITF Interval of Convergence of The Power Series Exam Practice

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10.1, 10.2 Sequences
Tuesday, June 22, 2021
6:02 PM
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10.3, 10.4 Series
Tuesday, June 29, 2021
6:04 PM
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10.5, 10.6 Comparison Tests, Alternating Series
Tuesday, July 6, 2021
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11.2 Power Series
Thursday, July 15, 2021
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11.3 Taylor, MacLaurin Series
Tuesday, July 20, 2021
6:03 PM
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CALCULUS – II (Su21) EXAM – IV
(all work must be clearly shown and explained step by step, otherwise it will be considered
as an app work, and no credit will be given. Give yourself enough time so that you can
upload your work. I will only accept the work under the assignment link, so don’t
email me your work.)
(𝑥−2)𝑛
1. Find the interval of convergence of the power series ∑∞
𝑛=1 4
√𝑛+3
4
2. Find the 6-th Taylor polynomial of 𝑓(𝑥) = √𝑥 centered at c = 1, and use the polynomial to
4
approximate √1.4.
4
3. Find a power series centered at c = 3 for 𝑔(𝑥) = 9−2𝑥
3
4. Find the MacLaurin series of 𝑔(𝑥) = 𝑒 2𝑥 then of 𝑓(𝑥) = 𝑒 −2𝑥 . Use the first six terms of the
3
1
last series to approximate ∫0 𝑒 −2𝑥 𝑑𝑥
5. Find the interval of convergence of the power series ∑∞
𝑛=1
(𝑥+1)𝑛
𝑛4 +2
𝑛!
6. Use the Ratio Test to determine the convergence of ∑∞
𝑛=0 𝑛𝑛
7. Find a Maclaurin series of 𝑔(𝑥) = 𝑐𝑜𝑠 𝑥 𝑡ℎ𝑒𝑛 𝑓𝑜𝑟 𝑐𝑜𝑠( 𝑥 5 ). Use the first five terms of
1
the series to approximate ∫0 𝑐𝑜𝑠( 𝑥 5 )𝑑𝑥
11
8. Find a power series centered at c = -4 for 𝑔(𝑥) = 16+3𝑥
𝑛
9. Use the Root Test to determine the convergence of: ∑∞
𝑛=1(1 − 7/𝑛) . If the test fails, use a
different one.
1
1
10. Find a power series representation for 𝑓(𝑥) = 1+𝑥 then for 𝑓(𝑥) = 1+𝑥 2 , and use the first
1
seven terms of the appropriate series to approximate ∫0 𝑎𝑟𝑐𝑡𝑎𝑛( 𝑥 4 )𝑑𝑥.
11. Find a Maclaurin series of 𝑔(𝑥) = 𝑠𝑖𝑛 𝑥 𝑡ℎ𝑒𝑛 𝑓𝑜𝑟 𝑠𝑖𝑛( 𝑥 6 ). Use the first five terms of
1
the series to approximate ∫0 𝑠𝑖𝑛( 𝑥 6 )𝑑𝑥
12. Find the 6-th Taylor polynomial of 𝑓(𝑥) = √𝑥 centered at c = 9, and use the polynomial to
approximate √9.5

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