`1+n/3+(n(n+1))/(3.6)+(n(n+1)(n+2))/(3. 6.9)+...oo=`

If n is a non zero rational number then show that 1 + n/2 + (n (n - 1))/(2.4) + (n(n-1)(n - 2))/(2.4.6) + ….. = 1 + n/3 + (n (n + 1))/(3.6) + (n (n + 1) (n + 2))/(3.6.9) + ….

If n is a non zero rational number then show that 1 + n/2 + (n (n - 1))/(2.4) + (n(n-1)(n - 2))/(2.4.6) + ….. = 1 + n/3 + (n (n + 1))/(3.6) + (n (n + 1) (n + 2))/(3.6.9) + ….

Prove that 5^(n) (1+(n)/(5) +(n(n-1))/(5*10) +(n(n-1)(n-2))/(5*10*15)+…oo)=3^(n) (1+(n)/(2)+(n(n+1))/(2*4)+(n(n+1)(n+2))/(2*4*6)+…oo)

lim_ (n rarr oo) ((1) / (n + 1) + (1) / (n + 2) + (1) / (n + 3) + ...... + (1) / (6n ))

lim_ (n rarr oo) [(1) / (n) + (n ^ (2)) / ((n + 1) ^ (3)) + (n ^ (2)) / ((n + 2) ^ (3)) + ...... + (1) / (8n)]

lim_ (n rarr oo) (1) / (n) [(1) / (n + 1) + (2) / (n + 2) + ... + (3n) / (4n)]

lim_ (n rarr oo) (3) / (n) [1 + sqrt ((n) / (n + 3)) + sqrt ((n) / (n + 6)) + sqrt ((n) / (n +9)) + ...... + sqrt ((n) / (n + 3 (n-1)))

Prove be mathematical induction : (1)/(3.6)+(1)/(6.9)+(1)/(9.12)+...+(1)/(3n(3n+3))=(n)/(9(n+1))

lim_ (n rarr oo) {(1) / (n ^ (3) +1) + (4) / (n ^ (3) +1) + (9) / (n ^ (3) +1) +. ........... + (n ^ (2)) / (n ^ (3) +1)}