For n in P, prove
n
Σ 1/√i = √n
i=1
Prove: Sum (i=1 to n) (1/√i) >= √n
By induction:
n=1, trivial. 1 = 1, true.
Assume this is true for n. Then:
Sum (i=1 to n+1) (1/√i) = Sum (i=1 to n) (1/√i) + 1/√n >= √n + 1/√n
= (n + 1) / √n > (n+1) / √(n+1) = √(n+1)
Sum (i=1 to n+1) (1/√i) >= √(n+1)
QED
Copyright © 2024 1QUIZZ.COM - All rights reserved.
Answers & Comments
Verified answer
Prove: Sum (i=1 to n) (1/√i) >= √n
By induction:
n=1, trivial. 1 = 1, true.
Assume this is true for n. Then:
Sum (i=1 to n+1) (1/√i) = Sum (i=1 to n) (1/√i) + 1/√n >= √n + 1/√n
= (n + 1) / √n > (n+1) / √(n+1) = √(n+1)
Sum (i=1 to n+1) (1/√i) >= √(n+1)
QED