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 3 n [ ( 1 + 3 i n ) 3 − 2 ( 1 + 3 i n ) ] and obtain the limit as follows. lim n → ∞ ∑ i = 1 n 3 n [ ( 1 + 3 i n ) 3 − 2 ( 1 + 3 i n ) ] = lim n → ∞ ∑ i = 1 n 3 n [ 1 + 9 i n + 27 i 2 n 2 + 27 i 3 n 3 − 2 − 6 i n ] = lim n → ∞ ∑ i = 1 n [ 81 i 3 n 4 + 81 i 2 n 3 + 9 i n 2 − 3 n ] = lim n → ∞ [ 81 n 4 ( n ( n + 1 ) 2 ) 2 + 81 n 3 ( n ( n + 1 ) ( 2 n + 1 ) 2 ) + 9 n 2 ( n ( n + 1 ) 2 ) − <

8th Edition

ISBN: 9781305266704

Author: James Stewart

Publisher: Brooks Cole

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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 46E

To determine

To find: The value of limn→∞∑i=1n3n[(1+3in)3−2(1+3in)].

Expert Solution & Answer

Chapter E Solutions

Single Variable Calculus

Ch. E - Prob. 1E Ch. E - Prob. 2E Ch. E - Write the sum in expanded form. 3. i=463i Ch. E - Prob. 4E Ch. E - Prob. 5E Ch. E - Write the sum in expanded form. 6. k=58xk Ch. E - Prob. 7E Ch. E - Write the sum in expanded form. 8. j=nn+3j2 Ch. E - Prob. 9E Ch. E - Prob. 10E

Ch. E - Prob. 11E Ch. E - Prob. 12E 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 - Prob. 26E Ch. E - Prob. 27E Ch. E - Prob. 28E Ch. E - Prob. 29E Ch. E - Prob. 30E Ch. E - Prob. 31E Ch. E - Prob. 32E Ch. E - Find the value of the sum. 33. i=1n(i+1)(i+2)Ch. E - Prob. 34E Ch. E - Prob. 35E Ch. E - Find the number n such that i=1ni=78.Ch. E - Prob. 37E Ch. E - Prove formula (e) of Theorem 3 using mathematical...Ch. E - Prove formula (e) of Theorem 3 using a method...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 - Prove the generalized triangle inequality:...Ch. E - Find the limit. 43. limni=1n1n(in)2 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]

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