[Algebra]
Problem: Let $T=32$. Given that $\sin \theta=\frac{\sqrt{T^{2}-64}}{T}$, compute the largest possible value of the infinite series $\cos \theta+\cos ^{2} \theta+\cos ^{3} \theta+\ldots$.

Solution: Using $\sin ^{2} \theta+\cos ^{2} \theta=1$ gives $\cos ^{2} \theta=\frac{64}{T^{2}}$, so to maximize the sum, take $\cos \theta=\frac{8}{|T|}$. Using the formula for the sum of an infinite geometric series gives $\frac{8 /|T|}{1-8 /|T|}=\frac{8}{|T|-8}$. With $T=32$, the answer is $\frac{8}{24}=\frac{1}{3}$.

Final answer: $\frac{1}{3}$