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sum(r=0)^(n)(-1)^(r)*^(n)C(r)(1+r*log(e)...

sum_(r=0)^(n)(-1)^(r)*^(n)C_(r)(1+r*log_(e)10)/((1+log_(e)10^(n))^(r))=

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If C_(0),C_(1),C_(2),.........,C_(3) denotes the binomial coeffcients in the expanssion of (1+x)^(a), then sum_(r rarr a)^(a)(-1)^(rn)C_(r).(1+r log_(e)10)/((1+log_(e)10^(x))^(r))=

sum_(r=0)^(n)(-1)^(r)*^(n)C_(r)(1+r ln10)/((1+ln10^(n))^(r))

sum_(r=0)^n (-1)^r .^nC_r (1+rln10)/(1+ln10^n)^r

Statement-1: sum_(r=0)^(n) (1)/(r+1) ""^(n)C_(r) = (1)/((n+1)x) {( 1 + x)^(n+1) -1}^(-1) Statement-2: sum_(r=0)^(n) (""^(n)C_(r))/(r+1) = (2^(n+1))/(n+1) .

Statement-1: sum_(r=0)^(n) (1)/(r+1) ""^(n)C_(r) = (1)/((n+1)x) {( 1 + x)^(n+1) -1}^(-1) Statement-2: sum_(r=0)^(n) (""^(n)C_(r))/(r+1) = (2^(n+1))/(n+1) .

Ifn>2than sum_(r=0)^(n)(-1)^(r)(n-r)(n-r+1)C_(r)=(A)0(B)n(C)2^(n)(D)(n-1)2^(n)

Statement -2: sum_(r=0)^(n) (-1)^( r) (""^(n)C_(r))/(r+1) = (1)/(n+1) Statement-2: sum_(r=0)^(n) (-1)^(r) (""^(n)C_(r))/(r+1) x^(r) = (1)/((n+1)x) { 1 - (1 - x)^(n+1)}

Statement -2: sum_(r=0)^(n) (-1)^( r) (""^(n)C_(r))/(r+1) = (1)/(n+1) Statement-2: sum_(r=0)^(n) (-1)^(r) (""^(n)C_(r))/(r+1) x^(r) = (1)/((n+1)x) { 1 - (1 - x)^(n+1)}