- #1
Aziza
- 190
- 1
Homework Statement
How do you prove that the binomial series (x+1)^p converges for |x|<1 ?
Homework Equations
The Attempt at a Solution
(x+1)^p = Ʃ[itex]x^{n}\frac{p!}{(p-n)!n!}[/itex]
After doing ratio test I get |x|<1 . But now I have to test end points and this is my problem:
when x=1,
an = [itex]\frac{p!}{(p-n)!n!}[/itex]
when x=-1,
an = (-1)^n[itex]\frac{p!}{(p-n)!n!}[/itex]
Since the interval of convergence does not include these endpoints, I know these two series must diverge, but how to prove this? Is it because as n->∞, (p-n)! eventually becomes undefined, and so the limit of an as n->∞ is undefined, but in order for the series to converge, an must approach 0 as n->∞ ?