Which of the series in Exercises 57–64 converge, and which diverge? Give reasons for your answers. 2"n!n! (-1)" (3n)! 57. У (2n)! 58. E n!(n + 1)!(n + 2)! n=1 n= (n!)" (n!)" 60. E(-1)". nln²) 59. n=1 (n")2 n=1 п" 61. У | 20*) п" 62. > n=1 (2")2 00 1.3 63. У · • (2n – 1) 4"2"n! n=1 1:3. ...•(2n – 1) 64. E [2•4....•(2n)](3" + 1) n=1

Q: Find the domain of the function. (Enter your answer using interval notation.) g(x) = V² - 6x

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Q: Question 7. Using induction prove that 13 + 23 +... + n³ = + u)u for n > 1.

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Q: 11. Prove by mathematical induction that > k (k – 1) = (n + 1)n(n – 1) 1sksn

A: To prove:∑1≤k≤nk(k−1)=13(n+1)(n)(n−1)Proposition:- For any n∈N∑1⩽k≤nk(k−1)=13(n+1)n(n−1)⇒ 1(1-1) +…

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Q: [n(n+1)]² Question 7. Using induction prove that 13 + 23 +...+ n3 for n > 1.

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Q: (3) 2.3-= 3" – 1 i=1 k+1 Hint: 2-3-1 = 2-3i-1+2-3'k+1)-1 = 3* –1+2-3* = 3-3* –1 = 3*+1_1 i=1 i=1

A: ∑i=1n2.3i-1=3n-1for  n=1∑i=1n2.3i-1=2.30=2=31-1,    truefor n=2∑i=1n2.3i-1=2.30+2.31=2+6=8=32-1,…

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Q: Find the sum of each series in Exercises 45–52. 4 6. 45. У 46. 2 3)(4n + 1) (4n (2n – 1)(2n + 1) n=1…

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