Let `S_(n)=1+(1)/(2)+(1)/(3)+….+(1)/(n)` Show that `s_(n)=n-((1)/(2)+(2)/(3))+…….+(n-1)/(n))`.

Let S_(n)=1+(1)/(2)+(1)/(3)+….+(1)/(n) Show that ns_(n)=n+((n-1)/(1)+(n-2)/(2)+………+(2)/(n-2)+(1)/(n-1)) .

Let S_(n)=1+(1)/(2)+(1)/(3)+(1)/(4)+......+(1)/(2^(n)-1) Then

Suppose a series of n terms given by S_(n)=t_(1)+t_(2)+t_(3)+ . . . . +t_(n) then S_(n-1)=t_(1)+t_(2)+t_(3)+ . . . . +t_(n-1),nge1 subtracting we get S_(n)-S_(n-1)=t_(n),nge2 surther if we put n=1 is the first sum then S_(1)=t_(1) thus w can write t_(n)=S_(n)-S_(n-1),nge2 and t_(1)=S_(1) Q. The sum of n terms of a series is a.2^(n)-b . where a and b are constant then the series is

If a_(1),a_(2),a_(3)….a_(n) are positive and (n-1)s = a_(1)+a_(2)+….+ a_(n) then prove that a_(1),a_(2),a_(3)…a_(n) ge (n-1)^(n)(s-a_(1))(s-a_(2))….(s-a_(n))

Let s_(n)=1+(1)/(3)+(1)/(3^(2))+…+(1)/(3^(n-1)) . The least value of n in N such that (3)/(2) -S_(n) lt (1)/(400) is

Given,s_(n)=1+q+q^(2)+....+q^(n),S_(n)=1+(q+1)/(2)+((q+1)/(2))^(2)+...+((q+1)/(2))^(n),q!=1 prove that ^(n+1)C_(1)+^(n+1)C_(2)s_(1)+^(n+1)C_(3)s_(2)+......+^(n+1)C_(n+1)s_(n)=2^(n)S_(n)

Let S_(n)=sum_(r=1)^(oo)(1)/(n^(r)) and sum_(n=1)^(k)(n-1)S_(n)=5050, then k=

Let S_n=1/1^2 + 1/2^2 + 1/3^2 +….. + 1/n^2 and T_n=2 -1/n , then :