document.write( "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! \n" ); document.write( "

Algebra.Com's Answer #815469 by ikleyn(52790) $\"\"$ $\"About$

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\n" ); document.write( "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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document.write( "Consider the sum\r\n" );
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document.write( "     =  +  +  + ... + .       (1)\r\n" );
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document.write( "You can write it EQUIVALENTLY in the form\r\n" );
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document.write( "     = .       (2)\r\n" );
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document.write( "This sum is the Riemann sum for the integral of the function  f(x) =   over the interval [0,1].\r\n" );
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document.write( "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),\r\n" );
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document.write( "where the primitive (\"antiderivative\")  function  F(x)  is   F(x) = ln(2+x).\r\n" );
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document.write( "This difference  F(1) - F(0)  is  ln(3) - ln(2) = .\r\n" );
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document.write( "THEREFORE,  lim   when n tends to infinity  is  .\r\n" );
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