Question 1184743: Help me find the limit of (1 + √2 + √3 +... + √(n-1) + √n)/n^(3/2) as n goes to +infinity. Thank you! Found 2 solutions by greenestamps, ikleyn:Answer by greenestamps(13203) (Show Source):
An excel spreadsheet appears to show that the limit approaches (VERY slowly) the value of 2/3.
I have no idea how to find that result algebraically; assuming that is what you are looking for, re-post your question specifying that you are looking for an way to get the result with formal mathematics instead of a spreadsheet.
You can put this solution on YOUR website! .
Help me find the limit of ( + + + . . . + + ) / n^(3/2) as n goes to +infinity. Thank you!
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Consider the sum
= 1/n^(3/2) + sqrt(2)/n^(3/2) + sqrt(3)/n^(3/2) + . . . + sqrt(n-1)/n^(3/2) + sqrt(n)/n^(3/2). (1)
You can write it EQUIVALENTLY in the form
= . (2)
This sum is the Riemann sum for the integral of the function f(x) = over the interval [0,1].
When n tends to infinity (n---> oo), the Riemann sum (2) tends to the integral, which is equal to the difference F(1) - F(0),
where the primitive ("antiderivative") function F(x) is F(x) = (2/3)*x^(3/2).
This difference F(1) - F(0) is - = .
THEREFORE, lim when n tends to infinity is .
