2nd January 2026

AMM Problem 12550 Solution

Analysis: An integral limit of the difference of trigonometric powers

In this article, I provide the solution to AMM Problem 12550. This article is being published a day after the AMM deadline!
This is part of my December 31st 2025 deadline, of which I solved 5/7 problems that I'm sharing here! This is the 4th such article.

This problem is all about the following limit:

lim⁡n→∞∫0π/2∣sin⁡nx−cos⁡nx∣1/n dx \lim_{n \to \infty} \int_{0}^{\pi/2} \left|\sin^{n} x - \cos^{n} x\right|^{1/n} \, dx

As well as the rate of convergence via:

lim⁡n→∞n2(L−∫0π/2∣sin⁡nx−cos⁡nx∣1/n dx) \lim_{n\to\infty} n^{2}\left( L-\int_{0}^{\pi/2} \left|\sin^{n} x - \cos^{n} x\right|^{1/n}\,dx \right)

Where LL is the first limit.
This was the longest problem of the 5, but also mostly just a long computation. But don't be fooled, this computation required a lot of intuition, for instance, knowing where all the mass in the 2nd limit came from, using symmetry, focusing down on the important part only, and clever bounds. This problem required surgical-level of computation!

I hope you enjoyed reading the article

Any feedback is greatly appreciated!

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