Probability of a ball ever being chosen from an urn 
There is one blue ball in an urn. Every time (and indefinitely), 5 red balls are added and immediately 1 random ball is withdrawn without being returned. What is the probability of the blue ball ever being withdrawn?

My idea was to define $A_k$ - {The blue ball is withdrawn at the $k$ step}, and then the solution is:
$\mathbb{P}(\bigcup_{k=1}^{\infty}A_k)=\sum_{k=1}^{\infty}\mathbb{P}(A_k).$
But calculating a general $\mathbb{P}(A_k)$ does not appear to be very simple (unless I'm missing something), so I'm thinking maybe there is a better way of approaching this?
Hints would be best.

HINT:
The probability that the blue ball is not withdrawn in $k$ draws
$P(R) = \frac{5}{6} \times \frac{9}{10} \times .. \frac{4k+1}{4k+2} = \prod \limits_{i=1}^k \frac{4i+1}{4i+2}$
The probability that the blue ball will get withdrawn in $k$ draws $P(B \leq k)= 1 - P(R)$
