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Maths-

General

Easy

Question

$L t subscript n not stretchy rightwards arrow straight infinity end subscript sum from n equals 1 to n of open square brackets fraction numerator 1 over denominator left parenthesis 2 n plus 1 right parenthesis left parenthesis 2 n plus 3 right parenthesis end fraction close square brackets$

6
1
$\frac{1}{6}$
0

hint Hint:

In the question we will first simplify the sequence then use the L hospital theorem to find the required value.

The correct answer is: $1 over 6$

$L t subscript n not stretchy rightwards arrow straight infinity end subscript sum from n equals 1 to n of open square brackets fraction numerator 1 over denominator left parenthesis 2 n plus 1 right parenthesis left parenthesis 2 n plus 3 right parenthesis end fraction close square brackets equals limit as n rightwards arrow infinity of fraction numerator 1 over denominator 3.5 end fraction plus fraction numerator 1 over denominator 5.7 end fraction plus fraction numerator 1 over denominator 7.9 end fraction plus....... plus fraction numerator 1 over denominator open parentheses 2 n plus 1 close parentheses left parenthesis 2 n plus 3 right parenthesis end fraction equals limit as n rightwards arrow infinity of 1 half open parentheses 1 third minus 1 fifth plus 1 fifth minus 1 over 7 plus 1 over 7 minus 1 over 9 plus........ plus fraction numerator 1 over denominator 2 n plus 1 end fraction minus fraction numerator 1 over denominator 2 n plus 3 end fraction close parentheses equals limit as n rightwards arrow infinity of 1 half open parentheses 1 third minus fraction numerator 1 over denominator 2 n plus 3 end fraction close parentheses equals 1 half limit as n rightwards arrow infinity of open parentheses fraction numerator 2 n plus 3 minus 3 over denominator 3 left parenthesis 2 n plus 3 right parenthesis end fraction close parentheses equals 1 half limit as n rightwards arrow infinity of open parentheses fraction numerator 2 n over denominator 6 n plus 9 end fraction close parentheses u sin g space L space h o s p i t a l space t h e o r e m comma space equals 1 half limit as n rightwards arrow infinity of open parentheses fraction numerator begin display style fraction numerator d blank over denominator d n end fraction end style 2 n over denominator begin display style fraction numerator d blank over denominator d n end fraction end style 6 n plus 9 end fraction close parentheses equals 1 half cross times 2 over 6 equals 1 over 6$

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