Let T=[C1−(1+12)C2+(1+12+13)C3−⋯+(−1)n−1(1+12+13+⋯+1n)Cn]
The rth term of the series is
T(r)=(−1)r−1(1+12+13+⋯+1r)Cr
Observe that T(r)=1∫0(−1)r−1 Cr(1+x+x2+⋯+xr−1)dx
So consider a series whose rth term is
T1(r), where
T1(r)=(−1)r−1 Cr(1+x+x2+⋯+xr−1)=(−1)r−1 Cr1−xr1−x=(−1)r Crx−1−(−1)rxr Crx−1
⇒n∑r=1T1(r)=1(x−1)[n∑r=1(−1)r Cr−n∑r=1(−1)r Crxr]=1x−1(1−1)n+11−x(1−x)n=(1−x)n−1
∴ Sum, S=1∫0(1−x)n−1dx=1n
∴100limn→200(S)=0.5