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What is the value of (-1)r nCr (1+r ln 10)/ (1+ ln (10n))r where n is a positivie integer???
do you mean the summation of this series ??
(-1)r nCr (1+r ln 10)/(1+ ln (10n))r
Denominator is constant. take it equal to k.
k = 1+ ln (10n)
(-1)r nCr (1+r ln 10)/kr
= (-1)r nCr*1/kr + ln10 *(-1)r nCr r/kr ..............(i)
Put summation.
Summation of (-1)r nCr*1/kr from r=0 to n = (1-1/k)n
Summation of (-1)r nCr r/kr = -n/k * (1-1/k)n-1
[This can be evaluated by expanding (1-x)n then differentiating both sides wrt x, then multiplying both sides by x and putting x=1/k]
Putting the above values in equation (i),
(1-1/k)n + ln10 * (-n/k * (1-1/k)n-1)
= (n.ln10)n/kn - n.ln10.(n.ln 10)n-1/kn
= 0
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