- #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,

a

_{n}= [itex]\frac{p!}{(p-n)!n!}[/itex]

when x=-1,

a

_{n}= (-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 a

_{n}as n->∞ is undefined, but in order for the series to converge, a

_{n}must approach 0 as n->∞ ?