`1/(1.3)+1/(3.5)+1/(5.7)+.... (n-1) terms`

(1)/(1.3) + (1)/(3.5) + (1)/(5.7) + …. (n-3) terms

lim_(n rarr oo){(1)/(1.3)+(1)/(3.5)+(1)/(5.7)+....+(1)/((2n+1)(2n+3 ))

underset(n to oo)lim {(1)/(1.3)+(1)/(3.5)+(1)/(5.7)+.....+(1)/((2n-1)(2n+1))}=

Statement -1: (1^(2))/(1.3)+(2^(2))/(3.5)+(3^(2))/(5.7)+ . . . .+(n^(2))/((2n-1)(2n+1))=(n(n+1))/(2(2n+1)) Statement -2: (1)/(1.3)+(1)/(3.5)+(1)/(5.7)+ . . . .+(1)/((2n-1)(2n+1))=(1)/(2n+1)

Statement -1: (1^(2))/(1.3)+(2^(2))/(3.5)+(3^(2))/(5.7)+ . . . .+(n^(2))/((2n-1)(2n+1))=(n(n+1))/(2(2n+1)) Statement -2: (1)/(1.3)+(1)/(3.5)+(1)/(5.7)+ . . . .+(1)/((2n-1)(2n+1))=(1)/(2n+1)

By the principle of mathematical induction prove that the following statements are true for all natural numbers 'n' (a) (1)/(1.3)+(1)/(3.5)+(1)/(5.7)+......+(1)/((2n-1)(2n+1)) =(n)/(2n+1) (b) (1)/(1.4)+(1)/(4.7)+(1)/(7.10)+......+(1)/((3n-2)(3n+1)) =(n)/(3n+1)

Find the sum of the series to n terms and to infinity : (1)/(1.3)+ (1)/(3.5) +(1)/(5.7) +(1)/(7.9)+...

Prove by the method of induction, (1)/( 1.3) + (1)/( 3.5) + (1)/( 5.7) + . . . + (1)/( (2n - 1)(2n + 1)) = (n)/(2 n +1)

(1)/(1 . 3)+(1)/(3 . 5)+(1) /(5 . 7)+.. to n terms =

Show that 1/1.3+1/3.5+1/5.7+..........+n "terms"=n/(2n+1)