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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

  1. 6
  2. 1
  3. 1 over 6
  4. 0

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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