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Question

Solve: limn1+3+6+........n(n+1)/2n3

A
12
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B
13
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C
16
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D
18
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Solution

The correct option is C 16
limn1+3+6+n(n+1)2n3

=limnnm=1m(m+1)2n3
=limnnm=1m2+m2n3
=12limnnm=1m2+nm=1mn3

It is known that:
nm=1m=n(n+1)2
nm=1m2=n(n+1)(2n+1)6

Hence,
12limnnm=1m2+nm=1mn3

=limn1(2n3)[n(n+1)2+n(n+1)(2n+1)6]

=limn1(2n3)[n(n+1)2]+limn1(2n3)[n(n+1)(2n+1)6]

In case of limn
if the degree of the polynomial in the numerator < degree of the denominator, the limit is 0.
If the degrees of the numerator and denominator are equal, the limit is the ratio of the leading terms.

Hence,
=limn1(2n3)[n(n+1)2]+limn1(2n3)[n(n+1)(2n+1)6]

=0+22×6
=16

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