# Calculus 3problem

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BYLL

Mathematics

calculus 2

All American Career College

## Description

Question #2 , #11, #12

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Math 104E Chapter 10 to 11.2 Test (Briggs) Name : _______________________ Use the blanks for your answers, when provided. Otherwise, BOX your answers. Don’t confuse sequences with series; read the questions carefully. Show your work. Correct answers with inadequate work shown receive only partial credit. Which of the following sequences (NOT series) {an} converge? If convergent, give the limit. If not convergent, state why. (5 points each) 1) { 6+4𝑛 3𝑛+2 } Converges: Yes / No If convergent, the limit is ____________________ 2) {3 − 4(−1)𝑛 } Converges: Yes / No If convergent, the limit is ____________________ 3) {3 𝑐𝑜𝑠 ( 4 5𝑛 8𝑛+5 4) {ln⁡ (4𝑛−3)} )} Converges: Yes / No If convergent, the limit is ____________________ Converges: Yes / No If convergent, the limit is ____________________ 1 of 6 BH ___ Full answer problems. (10 points each) Show whether each of the following series converge. DO NOT just plug in values for n; use the rules we have been studying. Give the exact value of the sum of the series. If you do not believe the series converges, or that you shouldn’t be able to get an exact sum, clearly explain why. 5a) ∑∞ 𝑛=1 ( 5b) ∑∞ 𝑛=1 ( 6) 6+4𝑛 ) The sum = ____________________ ) The sum = ____________________ 3𝑛+2 4𝑛 +5 5𝑛 4 4 𝑛 𝑛+1 ∑∞ 𝑛=2 ( − ) The sum = _______________________ 2 of 6 BH ___ Show in detail whether each series converges or not. 7) ∑∞ 𝑛=5 6−2cos⁡(𝑛) 𝑛2 −4𝑛 Converges: Yes / No Does it matter that this series started at n = 5? Yes / No Explain. 3 ∞ 2⁡𝑛 √𝑛 8) ∑𝑛=1 2 𝑛 +9 Converges: Yes / No 3 of 6 BH ___ 9) Use the integral test to show whether ∑∞ 𝑘=1 6𝑘 2 𝑘 3 +4 converges. Converges: Yes / No 2−3𝑛 𝑛 ∞ 10) Show in detail whether the series ∑𝑛=1 ( ) converges. 4𝑛 4 of 6 BH ___ 𝑛 ∞ (𝑥−3) 11) For the power series ∑𝑛=1 3 𝑛 : 𝑛 5 ⁡ The interval of convergence for this power series is _____________________. (Don’t forget to check the endpoints.) 12) Show how to find the first four non-zero terms of the Taylor polynomial for cos(x) at (This is NOT the series you may have memorized.) p(x) = + + a = π/4 + 5 of 6 BH ___ Extra Credit : Find the first four non-zero terms, AND the general term, for a power series for the following functions : First, write the power series for a geometric series: 1 1−𝑢 = 1 + + + +…+ +… (↑ general term here ↑) Use u = x4 to find the power series for: 1 1−𝑥 4 = 1 + + + +…+ +… (↑ general term here ↑) Multiply through by 2x, to get: ⁡ 2𝑥 1−𝑥 4 = + + + +…+ +… (↑ general term here ↑) Then use this result to find a power series for: tanh-1(x2) = + + + +…+ +… (↑ general term here ↑) Hint: 𝑑 𝑑𝑥 1 𝑑𝑤 (𝑡𝑎𝑛ℎ−1 𝑤) = 1−𝑤2 𝑑𝑥 6 of 6 BH ___
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Math 104E Chapter 10 to 11.2 Test (Briggs)

Name : _______________________

Don’t confuse sequences with series; read the questions carefully.
Which of the following sequences (NOT series) {an} converge?
If convergent, give the limit. If not convergent, state why. (5 points each)
1)

Converges: Yes / No If converg...

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