McLaurin Expansion of finite sum

[phanhoc]

Would you please find the McLaurin expansion of the following series to help me:
M
Ʃ Binomial(m + q - 1,q) [(a x)^q /((a x + b)^(m + q)]
q=0

where M , m ℂ N^+; a, b > 0;
MANY THANKS FOR YOUR HELP.

 

[mathman]

By brute force: (ax+b)^m+q can be expanded (binomial theorem) into a finite sum of powers of ax. Then rearrange double sum to keep like powers of ax together.

 

[phanhoc]

By brute force: (ax+b)^m+q can be expanded (binomial theorem) into a finite sum of powers of ax. Then rearrange double sum to keep like powers of ax together.

MANY THANKS FOR YOUR HELP.
HOWEVER, (ax+b)^(m+q) is the denominator of a fraction.
Could you please solve this again for me.