document.write( "Question 1183560: Does the series\r
\n" ); document.write( "\n" ); document.write( "(1/√(n^2+1^2) + 2/√(n^2+2^2) +...+(n-1)/√(n^2+(n-1)^2)+ n/√(n^2+n^2))/n\r
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Algebra.Com's Answer #813954 by ikleyn(52794) $\"\"$ $\"About$

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\n" ); document.write( "Does the series
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\n" ); document.write( "converge as n goes to infinity? If it does, what is the sum?
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\n" ); document.write( "\n" ); document.write( "            This problem is not for beginning Calculus students. \r
\n" ); document.write( "\n" ); document.write( "            It is for mature/advanced Calculus students.\r
\n" ); document.write( "\n" ); document.write( "            Therefore, I will give my solution without going into details, assuming that the reader has an adequate level.\r
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document.write( "The sum is\r\n" );
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document.write( "     = .      (1)\r\n" );
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document.write( "Each term under the sum symbol can be estimated this way\r\n" );
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document.write( "     <=  =    from the top,                   (2)\r\n" );
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document.write( "and\r\n" );
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document.write( "     >=  =   from the bottom,       (3)\r\n" );
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document.write( "or  \r\n" );
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document.write( "     <=  <= .      (4)\r\n" );
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document.write( "It gives, in turn, the following estimations for the entire sums  \r\n" );
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document.write( "     <=  <= .      (5)\r\n" );
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document.write( "Next, each sum    contains the sum of arithmetic progression  1 + 2 + 3 + . . . + n = ,\r\n" );
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document.write( "therefore,    =  = .\r\n" );
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document.write( "From this point, estimations (5) can be re-written\r\n" );
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document.write( "     <=  <= .      (6)\r\n" );
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document.write( "Thus the terms    are asymptotically between    and  .\r\n" );
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document.write( "        The sequence    is, actually,  monotonically DECREASING  sequence.\r\n" );
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document.write( "From the first glance, it seems to be unexpected statement, but it is true and it can be strictly established formally.\r\n" );
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document.write( "Therefore, of the two estimations (6) from the top and from the bottom, the only BOTTOM estimation makes sense for us now.\r\n" );
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document.write( "So, we have the monotonically decreasing sequence    limited by the value    from the bottom --- THEREFORE,\r\n" );
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document.write( "     +--------------------------------------------+\r\n" );
document.write( "     |   the sequence    is converged.         |\r\n" );
document.write( "     +--------------------------------------------+\r\n" );
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