There are a few ways to write this, but try looking at it like this.
Whew. And I would definitely take the ratio test from here.
The hyper geometric series :the infinite series
[1+ (ab/1!c)x+[a(a+1)b(b+1)/(2!c(c+1)]x^2+{[a(a+1)(a+2)b(b+1)(b+2)]/[3!c(C+1)(c+1)(c+2)]}x^3+....]
Where a,b and c are neither 0 nor negative integers,,
Please can anyone help me to prove by explaining that the hyper geometric series is converges??!
any help would be appreciated ,thank you
Ok. I'll try to Tex this all out.
Let or
So
All I did was substitute (n+1) for in, then divide that expression by a_n. But to simplify, I just flipped a_n and multiplied. Now for some cancellation.
For some reason, that Latex won't make that last bracket! Anyway, cancel some terms out and you'll get.
Now you have your common ratio! So take the limit as to see how the terms trend torwards infinity, and you should be able to show that this converges for all x.
Hopefully that makes sense.
I try to take limit as I get
And conclude that the series diverges for x>1 , converges for x<1 and test failed at x=1 .
We can use Gauss' test to see series behavior ..
But I have one question :
Is it necessary that we test series behavior at x=-1 or no??!
thanks