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MathCalculus

Antiderivative of 1/x: Evaluate ∫2/x dx

Evaluate OpenStax Calculus Volume 1 exercise 325 using the antiderivative of 1/x and explain why absolute value and x ≠ 0 matter.

MathCalculus

Alternating Series Test: Classify ∑ (−1)^(n+1)√(n+3)/n

Verify decreasing magnitudes and a zero limit, then compare the absolute terms with 1/√n to establish conditional convergence.

MathCalculus

Integral Test: Does ∑ 1/[n(ln n)²] Converge?

Check positivity, continuity, and monotonicity on [2,∞), then use u=ln x to prove convergence without equating the sum and integral.

MathCalculus

Alternating Series Error Bound: How Many Terms Approximate cos 1?

Count the constant term, bound the first omitted factorial term, and show why five terms meet the 0.00001 tolerance while four do not.

MathGeometry

Sierpinski Gasket: Find Three IFS Maps and an Orbit Distance Bound

Write the three half-scale affine maps, then use invariance and contraction to prove d(xₖ,G) ≤ 2⁻ᵏ for a starting point in the unit triangle.

MathCalculus

Divergence Theorem: Outward Flux Through a Capped Paraboloid

Use the divergence theorem to compute outward flux through a paraboloid and its base, then check the boundary orientation.

MathLinear Algebra

Scalar Product of Two Three-Dimensional Vectors

Find the scalar product of two three-dimensional vectors and use the zero result to check perpendicularity.

MathCalculus

Spherical Coordinate System: Convert the Point (-1, 2, 1)

Convert (-1, 2, 1) to spherical coordinates, resolve the quadrant, and round the angles to the nearest degree.

MathCalculus

Cylindrical Coordinate System: Convert a Shifted Cylinder Equation

Convert a shifted cylinder equation to cylindrical coordinates while preserving the r = 0 coordinate cases.