Convergence of Infinite Series

[Vagrant]

Homework Statement

1+ \frac{\alpha\beta}{\gamma} x + \frac{\alpha (\alpha+1)\beta(\beta+1)}{1.2.\gamma(\gamma+1)}x^{2}+...

Homework Equations

The Attempt at a Solution

Using D'Alembert's ratio test, I get lim_{n\rightarrow\infty}\frac{U_{n+1}}{U_{n}}=x
so, x>1 diverging series
x<1 converging series
when x=1, Using Raabe's test I get
lim_{n\rightarrow\infty}n[\frac{U_{n}}{U_{n+1}}-1]=\gamma-\alpha-\beta
so, Series Converges if \gamma-\alpha-\beta>1
and diverges if \gamma-\alpha-\beta<1

However the book has given the answer to be
coverges if \gamma-\alpha-\beta>0
and diverges if \gamma-\alpha-\beta<0

Can anyone point out my mistake, please?

 

[Gib Z]

http://mathworld.wolfram.com/HypergeometricFunction.html

Start reading at equation 7.

Edit: Btw, definitely not precalc math.

 

[Avodyne]

For Raabe's test wtih x=1, I get
\lim_{n\rightarrow\infty}n\!\left({\textstyle{U_{n}\over U_{n+1}}-1\right)=\gamma-\alpha-\beta+1

 

[Vagrant]

My expression for U_{n} = \frac{1.\alpha...(\alpha + n-1) 1.\beta...(\beta+n-1)}{1.2...(n-1) 1.\gamma...(\gamma+n-1)}

Using this I get \gamma-\alpha-\beta from Raabe's test