SOLUTION: Please help me find the limit of 1/(2n+1) + 1/(2n+2) + 1/(2n+3) + ... + 1/(3n) as n goes to +infinity. Thank you!

Question 1184792: Please help me find the limit of 1/(2n+1) + 1/(2n+2) + 1/(2n+3) + ... + 1/(3n) as n goes to +infinity. Thank you!
Answer by ikleyn(52786) (Show Source): You can put this solution on YOUR website!
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Please help me find the limit of 1/(2n+1) + 1/(2n+2) + 1/(2n+3) + ... + 1/(3n) as n goes to +infinity. Thank you!
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Consider the sum


     =  +  +  + ... + .       (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) = ln(2+x).



This difference  F(1) - F(0)  is  ln(3) - ln(2) = .



THEREFORE,  lim   when n tends to infinity  is  .

Solved.

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