`sum_(r=0)^n(-1)^r*^n C_r(1+rdotlog_e 10)/((1+log_e1 0^n)^r)=`

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

Let for n in N, f(n)=sum_(r=0)^(n)(-1)^(r)(C_(r)2^(r+1))/((r+1)(r+2))

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)}

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) .