Question 1103595
Sorry, not a closed form solution, but some values as n gets "large"…
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Let {{{ S[n] }}}  = Partial sum n,    {{{ n>= 1 }}}
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{{{ S[1] = 1/2 = 0.5 }}}
{{{ S[2] = 0.65 }}}
{{{ S[3] = 0.6974359 }}}
: : :
{{{ S[1000] = 0.7851481 }}}
: : :
{{{ S[10000] = 0.7853732 }}}
: :  :
{{{ S[100000] = 0.7853957 }}}
: : :
{{{ S[1000000] = 0.7853979 }}}
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Values found using a Perl one-liner  (if you have Perl installed, edit the value of $m to set the max value)
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perl -e '$m=1000000; for($i=0, $s=0; $i<$m; $i++) { $s += $m/($m*$m+($i+1)*($i+1)); }; print "Sum = $s\n";'
Sum = 0.785397913397392