Show that `x/((1+x))<1n(1+x)` for `x>0`

solve tan^(-1) ((1-x)/(1+x)) = 1/2 tan^(-1) x "for" x gt0

If P(1)=0a n d(d P(x))/(dx)gtP(x) , for all xge1 . Prove that P(x)>0 for all x>1.

Show that (x-1) is a factor of x ^(n)-1.

Show that x^(n) =1 + n(1 -1/x) + (n(n+1))/1.2 (1 -1/x)^(2) + ...

Prove that (1+x)^(n) ge (1+nx) for all natural number n where x gt -1

Solve the tan^(-1)((1-x)/(1+x))=(1)/(2)tan^(-1)x for x gt0 .

In the neighbourhood of x=0 it is known that 1+|x|lt(e^(x)-1)/(x)lt1-|x|"then find"lim_(xto0)(e^(x)-1)/(x).

Prove that int_(0)^(1)(dx)/(1+x^(n))gt1-(1)/(n)"for n"inN