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Question

The value of
100limn200[C1(1+12)C2+(1+12+13)C3+(1)n1(1+12+13++1n)Cn] is (where Cr=nCr)

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Solution

Let T=[C1(1+12)C2+(1+12+13)C3+(1)n1(1+12+13++1n)Cn]

The rth term of the series is
T(r)=(1)r1(1+12+13++1r)Cr

Observe that T(r)=10(1)r1 Cr(1+x+x2++xr1)dx

So consider a series whose rth term is
T1(r), where
T1(r)=(1)r1 Cr(1+x+x2++xr1)=(1)r1 Cr1xr1x=(1)r Crx1(1)rxr Crx1

nr=1T1(r)=1(x1)[nr=1(1)r Crnr=1(1)r Crxr]=1x1(11)n+11x(1x)n=(1x)n1
Sum, S=10(1x)n1dx=1n
100limn200(S)=0.5

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