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Question

If Sn denotes the sum to n terms of the series
cot−174+cot−1194+cot−1394+cot−1674+.... then

A
Sn=tan1n2n+5
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B
Sn=cot1n+52n
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C
Sn=tan14n2n+5
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D
limnSn=cot112
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Solution

The correct option is C Sn=tan14n2n+5
cot174+cot1194+cot1394+........
=cot1(4+34)+cot1(16+34)+cot1(30+34)+......
=cot1(1+34)+cot1(4+34)+cot1(9+34)+....
=nr=1cot1(r2+34)
=rr=1tan1(1r2+3/4)=tan1(11+(r21/4))
=tan1(11+(r+1/2)(r1/2))=tan1((r+1/2)(r1/2)1+(r+1/2)(r1/2))
=nr=1tan1(r+1/2)tan1(r1/2)
=tan1(n+12)
=tan1(n+12)(12)
=tan1⎜ ⎜ ⎜ ⎜n+112121+12(n+12)⎟ ⎟ ⎟ ⎟=tan1⎜ ⎜ ⎜n1+n2+14⎟ ⎟ ⎟
=tan1⎜ ⎜ ⎜nn2+54⎟ ⎟ ⎟=tan1(4n5+2n)


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