सिद्ध कीजिए की :
` (1)/(n+1) +(1)/( 2(n +1) ^(2))+ ( 1)/(3( n+1) ^(3))+..........=( 1)/(n) -(1)/( 2n ^(2) )+ ( 1)/( 3n ^(3))-..........`

If (1)/(n+1)+(1)/(2(n+1)^(2))+(1)/(3(n+1)^(3))+….= lambda((1)/(n)-(1)/(2n^(2))+(1)/(3n^(3))-……) then lambda=

If (1)/(n+1)+(1)/(2(n+1)^(2))+(1)/(3(n+1)^(3))+….= lambda((1)/(n)-(1)/(2n^(2))+(1)/(3n^(3))-……) then lambda=

lim_ (n rarr oo) (1+ (1) / (2) + (1) / (2 ^ (2)) + (1) / (2 ^ (3)) + ...... (1) / (2 ^ (n))) / (1+ (1) / (3) + (1) / (3 ^ (2)) + (1) / (3 ^ (3)) ...... (1) / (3 ^ (n)))

Lt_(ntooo)[(1)/(3n+1)+(1)/(3n+2)+............+(1)/(3n+n)]=

1.3+3.5+5.7+......+(2n-1)(2n+1)=(n(4n^(2)+6n-1))/(3)

1.2+2.3+3.4+.........+n(n+1)=(1)/(3)n(n+1)(n+3)

(1)/(1.2)+(1)/(2.3)+(1)/(3.4)+.......+(1)/(n(n+1))=(n)/(n+1),n in N is true for

Prove that (1)/(n+1)=(nC_(1))/(2)-(2(^(n)C_(2)))/(3)+(3(^(n)C_(3)))/(4)-...+(-1)^(n+1)(n(^(n)C_(n)))/(n+1)