The Basel problem
Eight thousand terms. The missing tail is about 1/n, and the gap is that size.
Open in the documentationExamples
Series, products, and piecewise formulas. Each one opens in the documentation, where it can be solved.
Eight thousand terms. The missing tail is about 1/n, and the gap is that size.
Open in the documentationTwenty terms of each arctangent. The gap against π is one unit in the last place.
Open in the documentationThe 1/(2n) correction. At twenty thousand terms this is γ to about ten digits.
Open in the documentationForty factors. The product climbs toward π from below.
Open in the documentationC(n, 0) is an empty product, so it is 1. The two ways of writing (1 + x)^n agree exactly.
Open in the documentationThe probabilities add to 1. The mean is n p.
Open in the documentationThe first eight primes. What is missing is the rest of the primes, about one percent of π²/6.
Open in the documentationAt a sample point the result is the sample, so y is 3. The branch j = i is 1, and the division by zero is never evaluated.
Open in the documentationThe series uses only the odd harmonics. Near the jump the overshoot stays, which is the Gibbs phenomenon. Solve it to draw the chart.
Open in the documentationEach entry is an outer product, summed. The exact solution is the integer vector. d is roundoff: this matrix is badly conditioned.
Open in the documentationZ is the zero matrix. A 2×2 matrix satisfies its own characteristic equation. The trace and the determinant are read by index.
Open in the documentationThe mean of 1, 2, 3, 4 is 2.5. Newton returns the square root of 2. The ranged sum is 30 and the product is 15.
Open in the documentationy is −4. z is 64, because 3! is 6 and 2^6 is 64.
Open in the documentationWriting 2 and 3 next to each other multiplies them. y is 6. r is 1.
Open in the documentationx is the positive square root. The inequality is the constraint.
Open in the documentationThe largest value is 0, at a = 2.
Open in the documentationy is 0, z is 2, w is 3, s is 2ʝ, and n is 120.
Open in the documentationRejected. It is not −∞.
Open in the documentationy is 3. The else branch is not evaluated.
Open in the documentationy is 1. An arm after else is a syntax error.
Open in the documentationThe area under y = x from 0 to 1 is 1/2.
Open in the documentationThe derivative of x^2 at 3 is 6.
Open in the documentationThe Gaussian integral is √π.
Open in the documentation∞ is a number. 1/∞ is 0, and exp(−∞) is 0.
Open in the documentation∫ and integ are the same call. Both are 1/2.
Open in the documentationThe italic e is exp. The plain letter e stays 4.
Open in the documentationπ is the constant. ʝ is the imaginary unit.
Open in the documentation