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 ge 1 is a positive integer, then prove that 3^(n) ge 2^(n) + n . 6^((n - 1)/(2))

The sum to n terms of the series 1+ 5 ((4n +1)/(4n - 3)) + 9 ((4n +1) /( 4n -3)) ^(2) + 13 ((4n +1)/( 4n -3)) ^(3) + ((4n + 1 )/( 4n -3)) ^(2)+ ……. is .

1+n. (2n)/(1+n)+(n(n+1))/(1.2) ((2n)/(1+n))^2+…..

Prove that (.^(n)C_(1))/(2) + (.^(n)C_(3))/(4) + (.^(n)C_(5))/(6) + "…." = (2^(n) - 1)/(n+1) .

Prove that (.^(n)C_(1))/(2) + (.^(n)C_(3))/(4) + (.^(n)C_(5))/(6) + "…." = (2^(n) - 1)/(n+1) .

Find the sum of the series: 1. n+2.(n-1)+3.(n-2)++(n-1). 2+n .1.

Find the sum of the series: 1. n+2.(n-1)+3.(n-2)++(n-1). 2+n .1.

If ((2n+1),(0))+ ((2n+1),(3)) +((2n+1),(6))+ ...= 170, then n equals

Using the principle of mathematical induction, prove that : 1. 2. 3+2. 3. 4++n(n+1)(n+2)=(n(n+1)(n+2)(n+3))/4^ for all n in N .

Let n is a rational number and x is a real number such that |x|lt1, then (1+x)^(n)=1+nx+(n(n-1)x^(2))/(2!)+(n(n-1)(n-2))/(3!).x^(3)+ . . . This can be used to find the sum of different series. Q. Sum of infinite series 1+(2)/(3)*(1)/(2)+(2)/(3)*(5)/(6)*(1)/(2^(2))+(2)/(3)*(5)/(6)*(8)/(9)*(1)/(2^(3))+ . . .oo is