You’re not wrong that your formula computes to 2 regardless what λ is. The difference is, this formula is trivial. Another difference is, the formula over there

You’re not wrong that your formula computes to 2 regardless what λ is. The difference is, this formula is trivial. Another difference is, the formula over there is an infinite series. What it means is, first, try prove that formula does not depends on λ. It is not obvious on first glance it is. (You may try to d/dλ and proof that equals 0.) Second, change λ, it gives you a different infinite series, while each converge to π, they converge at a different rate. In fact, part of the game of computing digits of π is to use different formula, where each converges to π eventually, but they have different converging “speed”, in the sense that giving n=100 terms, some gives you much more accurate digits. Also, since each term has a different computing “speed” too, ie using a computer to compute a term might be much slower in one series, that even if it has better converging “speed”, would be very slow to compute. The art of breaking world record then is to find a balance between the 2, such that overall it is fastest. Regardless of that though, discovering a whole family of series converging to π is in itself interesting. A special case reducing it to something we already know is not too surprising, but a good entry point for a story.