InfiniteMath

All equations

• A Sum of Squares: \sum_{k=1}^{5} k^{2} = \frac{5\cdot6\cdot11}{6} = 55 — A closed form for the square pyramid of 5 layers.
• A Finite Geometric Sum: \sum_{k=0}^{4} 2^{\,k} = \frac{2^{5} - 1}{1} = 31 — Powers of 2 up to the 4th, folded into one quotient.
• Symmetry in Pascal: \binom{3}{1} = \binom{3}{2} = 3 — Choosing 1 of 3 is the same as leaving 2 behind.
• A Continued Fraction: \sqrt{50} = 7 + \cfrac{1}{2\cdot7 + \cfrac{1}{2\cdot7 + \cdots}} — Irrational square roots unfold into endlessly repeating patterns of integers.
• A Continued Fraction: \sqrt{10} = 3 + \cfrac{1}{2\cdot3 + \cfrac{1}{2\cdot3 + \cdots}} — Irrational square roots unfold into endlessly repeating patterns of integers.
• A Finite Geometric Sum: \sum_{k=0}^{4} 2^{\,k} = \frac{2^{5} - 1}{1} = 31 — Powers of 2 up to the 4th, folded into one quotient.
• A Sum of Squares: \sum_{k=1}^{5} k^{2} = \frac{5\cdot6\cdot11}{6} = 55 — A closed form for the square pyramid of 5 layers.
• A Sum of Squares: \sum_{k=1}^{9} k^{2} = \frac{9\cdot10\cdot19}{6} = 285 — A closed form for the square pyramid of 9 layers.
• Symmetry in Pascal: \binom{8}{7} = \binom{8}{1} = 8 — Choosing 7 of 8 is the same as leaving 1 behind.
• A Row of Pascal's Triangle: \sum_{k=0}^{6} \binom{6}{k} = 2^{6} = 64 — Every subset of a set of 6 things, counted by size and then all together.
• A Finite Geometric Sum: \sum_{k=0}^{5} 4^{\,k} = \frac{4^{6} - 1}{3} = 1365 — Powers of 4 up to the 5th, folded into one quotient.
• A Telescoping Sum: \sum_{k=1}^{11} \frac{1}{k(k+1)} = \frac{11}{12} — Almost every term cancels its neighbour; only the first and last survive.
• One Equals Point-Nine-Repeating: 1 = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots = 0.\overline{9} — Not an approximation — an exact equality, and the geometric series that proves it.
• Zeta of Four: \sum_{n=1}^{\infty} \frac{1}{n^{4}} = \frac{\pi^{4}}{90} — The Basel problem's bigger sibling. Every even value of the zeta function hides a power of π.
• De Moivre's Theorem: (\cos\theta + i\sin\theta)^{n} = \cos n\theta + i\sin n\theta — Raising a rotation to a power just multiplies the angle — trigonometry falling out of complex arithmetic.
• The Euler–Mascheroni Constant: \gamma = \lim_{n\to\infty}\left(\sum_{k=1}^{n} \frac{1}{k} - \ln n\right) — The gap between the harmonic series and the natural log. We still don't know if γ is irrational.
• The Series for e: e = \sum_{n=0}^{\infty} \frac{1}{n!} — Euler's number as the sum of reciprocal factorials — the fastest classic series there is.
• The Wallis Product: \frac{\pi}{2} = \prod_{n=1}^{\infty} \frac{4n^{2}}{4n^{2} - 1} — π built from an infinite product of rational numbers, discovered by John Wallis in 1656.
• Nicomachus's Theorem: \left(\sum_{k=1}^{n} k\right)^{2} = \sum_{k=1}^{n} k^{3} — The sum of the first n cubes is the square of the sum of the first n integers. A small miracle.
• Binet's Formula: F_{n} = \frac{\varphi^{n} - \psi^{n}}{\sqrt{5}} — The integer Fibonacci numbers, written purely with the irrational golden ratio φ and its conjugate ψ.
• Stirling's Approximation: n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^{n} — How factorials grow. Again π and e appear where you'd never expect a circle or growth constant.
• The Leibniz Series: \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots — π, from nothing but the odd numbers and alternating signs. Beautiful, and famously slow to converge.
• Euler's Product: \zeta(s) = \prod_{p\ \text{prime}} \frac{1}{1 - p^{-s}} — A sum over all integers equals a product over only the primes — the seed of analytic number theory.
• The Golden Ratio: \varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cdots}}} — The 'most irrational' number: the slowest continued fraction to converge, made only of 1s.
• Euler's Formula: e^{ix} = \cos x + i\sin x — The bridge between exponential growth and rotation — the reason complex numbers describe waves.
• The Gaussian Integral: \int_{-\infty}^{\infty} e^{-x^{2}}\,dx = \sqrt{\pi} — The area under the bell curve. There's no elementary antiderivative, yet the total area is exactly √π.
• The Basel Problem: \sum_{n=1}^{\infty} \frac{1}{n^{2}} = \frac{\pi^{2}}{6} — Euler's 1734 result: the reciprocals of the squares sum to π²/6. Nobody expected π to show up here.
• Euler's Identity: e^{i\pi} + 1 = 0 — Five of the most important constants — e, i, π, 1, and 0 — bound together in one line. Often called the most beautiful equation in mathematics.