If a>1 and ⁿ√a = 1+x, prove that 0 < x < a/n

[samii]

if a>1 and ⁿ√a = 1 + x, prove that 0 < x < a/n
Deduce that ⁿ√a →1, n →∞
confused!

 

[tiny-tim]

hi samii!

rewrite ⁿ√a = 1 + x

 

[HallsofIvy]

If $\sqrt[n]{a}= 1+ x$, then $a= (1+ x)^n$. Expand using the binomial theorem to get that a= 1+ nx+ positive terms so that 1+ nx< a. Since x< a/n, as n goes to infinity, x goes to 0.

 

[samii]

hii again :)
thanks also if 0 < a < 1
what is the corresponding result?

 

[tiny-tim]

then ⁿ√a would be < 1, so you'd have to write ⁿ√a = 1 - x …

so what would happen then? ​