Explanation Theorem used: Let c be a constant and n be a positive integer. Then, ∑ i = 1 n c = n c , ∑ i = 1 n i = n ( n + 1 ) 2 and ∑ i = 1 n i 2 = n ( n + 1 ) ( 2 n + 1 ) 6 . Calculation: Simplify the expression lim n → ∞ ∑ i = 1 n 2 n [ ( 2 i n ) 3 + 5 ( 2 i n ) ] and obtain the limit as follows. lim n → ∞ ∑ i = 1 n 2 n [ ( 2 i n ) 3 + 5 ( 2 i n ) ] = lim n → ∞ ∑ i = 1 n 2 n [ 8 i 3 n 3 + 10 i n ] = lim n → ∞ ∑ i = 1 n [ 16 i 3 n 4 + 20 i n 2 ] = lim n → ∞ [ 16 n 4 ∑ i = 1 n i 3 + 20 n 2 ∑ i = 1 n i </

Single Variable Calculus: Early Transcendentals

8th Edition

ISBN: 9781305270336

Author: James Stewart

Publisher: Cengage Learning

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Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0…

Question

Chapter E, Problem 45E

To determine

To find: The value of limn→∞∑i=1n2n[(2in)3+5(2in)].

Expert Solution & Answer

Chapter E Solutions

Single Variable Calculus: Early Transcendentals

Ch. E - Prob. 1E Ch. E - Write the sum in expanded form. 2. i=161i+1 Ch. E - Prob. 3E Ch. E - Write the sum in expanded form. 4. i=46i3 Ch. E - Prob. 5E Ch. E - Prob. 6E Ch. E - Prob. 7E Ch. E - Prob. 8E Ch. E - Write the sum in expanded form. 9. j=0n1(1)j Ch. E - Write the sum in expanded form. 10. i=1nf(xi)xi

Ch. E - Prob. 11E Ch. E - Write the sum in sigma notation. 12. 3+4+5+6+7 Ch. E - Prob. 13E Ch. E - Write the sum in sigma notation. 14....Ch. E - Prob. 15E Ch. E - Prob. 16E Ch. E - Prob. 17E Ch. E - Prob. 18E Ch. E - Prob. 19E Ch. E - Prob. 20E Ch. E - Prob. 21E Ch. E - Prob. 22E Ch. E - Prob. 23E Ch. E - Prob. 24E Ch. E - Prob. 25E Ch. E - Find the value of the sum. 26. i=11004 Ch. E - Prob. 27E Ch. E - Prob. 28E Ch. E - Prob. 29E Ch. E - Prob. 30E Ch. E - Find the value of the sum. 31. i=1n(i2+3i+4)Ch. E - Prob. 32E Ch. E - Prob. 33E Ch. E - Prob. 34E Ch. E - Prob. 35E Ch. E - Prob. 36E Ch. E - Prob. 37E Ch. E - Prob. 38E Ch. E - Prob. 39E Ch. E - Prove formula (e) of Theorem 3 using the following...Ch. E - Evaluate each telescoping sum. (a) i=1n[i4(i1)4]...Ch. E - Prob. 42E Ch. E - Prob. 43E Ch. E - Prob. 44E Ch. E - Prob. 45E Ch. E - Prob. 46E Ch. E - Prob. 47E Ch. E - Prob. 48E Ch. E - Prob. 49E Ch. E - Prob. 50E

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Find the limit if it exists of Anx1 as n approaches infinity, where A=[02120], and x1=[aa]
Alligator Teeth Researchers have developed a mathematical model that can be used to estimate the number of teeth N(t) at time t days of incubation for Alligator mississippiensis, where N(t)=71.8e8.96e0.0685t. Source: Journal of Theoretical Biology. a. Find N(65), the number of teeth of an alligator that hatched after 65 days. b. Find limtN(t) and use this value as an estimate of the number of teeth of newborn alligator. Hint: See Exercise 71. Does this estimate differ significantly from the estimate of part a?

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