Winning at tennis What is your probability of winning a game of tennis, starting from the even score Deuce(40-40), if your probability of winning each point is 0.3 and your opponent's is 0.7?

My answer:
I think the sequence of independent trials are required to win a game of tennis starting from even score Deuce(40-40),each of which is a success with  probability (0.3 × 0.3 =0.09) or a failure with probability (0.7 × 0.3 =0.42). Suppose, the independent trials to win a game  of tennis are n. That means after (n-1) trials of failures, the nth trial is success.
$$ \displaystyle\sum_{n=0}^{\infty} 0.09 \cdot (0.42)^n= \lim_{n\to \infty}\frac{0.09\cdot(1-0.42^{n})}{1-021}=  0.15517 $$ 15.5%
Is this answer correct? If wrong, why?

This can be most easily solved using a recurrence formula. Consider the next two games. There are three possible outcomes. You either win (with probability $0.3 \times 0.3$), or lose (with probability $0.7 \times 0.7$) or get back to duece again (with probability $0.3 \times 0.7 + 0.7 \times 0.3$). By the law of total probability
$$P({\rm win}|{\rm duece})=1 \times 0.3 \times 0.3 + 0 \times 0.7 \times 0.7 + P({\rm win}|{\rm duece}) \times (0.3 \times 0.7 + 0.7 \times 0.3).$$
You can solve this formula for the required probability.
