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 #813957 by robertb(5830) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"lim%28n-%3Einfinity%2C+%281%2Fn%29%2Asum%28k%2Fsqrt%28n%5E2%2Bk%5E2%29%2C+k=1%2Cn%29%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "The term inside the summation eerily seems like the area of a rectangle whose width is $\"1%2Fn\"$ , and whose height is $\"%28k%2Fn%29%2Fsqrt%281%2B%28k%2Fn%29%29%5E2\"$ .
\n" ); document.write( "As such, this looks like the equipartition of the interval [0,1] in the construction of the upper Riemann sums for the function $\"f%28x%29+=+x%2Fsqrt%281%2Bx%5E2%29\"$ . Hence,\r
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\n" ); document.write( "\n" ); document.write( "The integral is easily evaluated as $\"sqrt%282%29-1\"$ . Therefore, the infinite series converges and the sum is $\"sqrt%282%29+-+1\"$ , approximately 0.41421 to 5 d.p.\r
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