(4-(1)/(n))+(4-(2)/(n))+(4-(3)/(n))+....

For n in N , (4-(2)/(1))(4-(2)/(2))(4-(2)/(3))(4-(2)/(4)).........(4-(2)/(n)) is

4C_(0)+(4^(2))/(2)*c_(1)+(4^(3))/(3)c_(2)+............+(4^(n+1))/(n+1)C_(n)=(5^(n+1)-1)/(n+1)

Prove the following by using the principle of mathematical induction for all n in N :- 1/(1.2.3)+1/(2.3.4)+1/(3.4.5)+...+1/(n(n+1)(n+2))=(n(n+3))/(4(n+1)(n+2)) .

lm_ (n rarr oo) ((1 ^ (3)) / (n ^ (4)) + (2 ^ (4)) / (n ^ (4)) + (3 ^ (3)) / (n ^ (4)) + ...... + (n ^ (3)) / (n ^ (4)))

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

lim_(n rarr oo)(((n+1)^(1/3))/(n^(4/3))+((n+2)^(1/3))/(n^(4/3))+.....+((2n)^(1/3))/(n^(4/3))) is equal to

4^(n)C_(0)+4^(2)(nC_(1))/(3)+4^(3)(nC_(2))/(3)+4^(4)(nC_(3))/(4)+....+4^(n+1)(nC_(n))/(n+1) is equal to