Tính tổng S = 1.2 + 2.3+.....+99.100

Tình tổng S = \(1+3+5+....+99\) 

Tính lim Un , biết :

a) \(\left\{{}\begin{matrix}U_1=\sqrt{2}\\U_{n+1}=\sqrt{2+U_n}\end{matrix}\right.\) , n \(\ge\) 1

b) \(\left\{{}\begin{matrix}U_1=\dfrac{1}{2}\\U_{n+1}=\dfrac{1}{2-U_n}\end{matrix}\right.\)  .

Tính

lim \(\left(\dfrac{1}{\sqrt{n^3+1}}+\dfrac{1}{\sqrt{n^3+2}}+....+\dfrac{1}{\sqrt{n^3+n}}\right)\)

Tính

lim \(\dfrac{2.1^2+3.2^2+......+\left(n+1\right)n^2}{n^4}\)

Tính

lim \(\left(\dfrac{1}{1.3}+\dfrac{1}{3.5}+.....+\dfrac{1}{\left(2n-1\right)\left(2n+1\right)}\right)\) 

Tính

lim \(\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+.....+\dfrac{1}{n\left(n+1\right)}\right)\)