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Question

$stack lim with n rightwards arrow infinity below invisible function application not stretchy sum from r equals 1 to n of fraction numerator r over denominator 1 cross times 3 cross times 5 cross times 7 cross times 9 cross times midline horizontal ellipsis cross times left parenthesis 2 r plus 1 right parenthesis end fraction blank$ is equal to

$\frac{1}{3}$
$\frac{3}{2}$
$\frac{1}{2}$
None of these

The correct answer is: $fraction numerator 1 over denominator 2 end fraction$

$T open parentheses r close parentheses equals fraction numerator r over denominator 1 cross times 3 cross times 5 cross times midline horizontal ellipsis cross times left parenthesis 2 r plus 1 right parenthesis end fraction$
$equals fraction numerator 2 r plus 1 minus 1 over denominator 2 left parenthesis 1 cross times 3 cross times 5 midline horizontal ellipsis open parentheses 2 r plus 1 close parentheses right parenthesis end fraction$
$equals fraction numerator 1 over denominator 2 end fraction blank open parentheses fraction numerator 1 over denominator 1 cross times 3 cross times 5 midline horizontal ellipsis open parentheses 2 r minus 1 close parentheses end fraction minus fraction numerator 1 over denominator 1 cross times 3 cross times 5 midline horizontal ellipsis open parentheses 2 r plus 1 close parentheses end fraction close parentheses$
$equals negative fraction numerator 1 over denominator 2 end fraction left square bracket V open parentheses r close parentheses minus V left parenthesis r minus 1 right parenthesis right square bracket$
$rightwards double arrow blank not stretchy sum from r equals 1 to n of T left parenthesis r right parenthesis equals negative fraction numerator 1 over denominator 2 end fraction left parenthesis V open parentheses n close parentheses minus V left parenthesis 0 right parenthesis right parenthesis$
$equals fraction numerator 1 over denominator 2 end fraction open parentheses 1 minus fraction numerator 1 over denominator 1 cross times 3 cross times 5 cross times midline horizontal ellipsis cross times open parentheses 2 n plus 1 close parentheses end fraction close parentheses$
$rightwards double arrow blank stack lim with n rightwards arrow infinity below invisible function application not stretchy sum from r equals 1 to n of fraction numerator r over denominator 1 cross times 3 cross times 5 cross times 7 cross times 9 cross times midline horizontal ellipsis cross times open parentheses 2 r plus 1 close parentheses end fraction$
$equals stack lim with n rightwards arrow infinity below fraction numerator 1 over denominator 2 end fraction invisible function application open parentheses 1 minus fraction numerator 1 over denominator 1 cross times 3 cross times 5 cross times midline horizontal ellipsis cross times open parentheses 2 n plus 1 close parentheses end fraction close parentheses equals fraction numerator 1 over denominator 2 end fraction$

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