Explanation Consider the sum ∑ i = 1 n [ ( 1 + i ) 4 − i 4 ] . ∑ i = 1 n [ ( 1 + i ) 4 − i 4 ] = ( 2 4 − 1 4 ) + ( 3 4 − 2 4 ) + ( 4 4 − 3 4 ) + ⋯ + ( ( n + 1 ) 4 − n 4 ) = ( n + 1 ) 4 − 1 4 = n 4 + 4 n 3 + 6 n 2 + 4 n + 1 − 1 = n 4 + 4 n 3 + 6 n 2 + 4 n Thus, ∑ i = 1 n [ ( 1 + i ) 3 − i 3 ] = n 4 + 4 n 3 + 6 n 2 + 4 n (1) Simplify the sum ∑ i = 1 n [ ( 1 + i ) 4 − i 4 ] by another method. ∑ i = 1 n [ ( 1 + i ) 4 − i 4 ] = ∑ i = 1 n [ ( i 4 + 4 i 3 + 6 i 2 + 4 i + 1 ) − i 4 ] = ∑ i = 1 n [ ( i 4 + 4 i 3 + 6 i 2 + 4 i + 1 ) − i 4 ] = ∑ i = 1 n 4 i 3 + 6 i 2 + 4 i + 1 = 4 ∑ i = 1 n i 3 + 6 ∑ i = 1 n i 2 + 4 ∑ i = 1 n i + ∑ i = 1 n 1 Further simplified as,

Single Variable Calculus: Early Transcendentals

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ISBN: 9781305270336

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Publisher: Cengage Learning

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Chapter E, Problem 39E

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To prove: The formula ∑i=1ni3=[n(n+1)2]2.

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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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To prove: The formula ∑ i = 1 n i 3 = [ n ( n + 1 ) 2 ] 2 .

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To prove: The formula ∑i=1ni3=[n(n+1)2]2.

Chapter E Solutions