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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???

9 years ago

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/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