Closed form

• How to Find Closed Forms for Summations
  Learn practical ways to find finite-sum formulas, including pattern recognition, telescoping, index shifts, and a worked example.
• How Many Ways Can You Climb n Stairs Using 2 or 3 at a Time?
  Count ways to climb n stairs using only 2- or 3-stair moves: derive the recurrence, generating function, and an explicit binomial-sum formula.
• How to Find a Closed Form for an nth Term Using Generating Functions
  Turn a recurrence into a generating function, simplify it, and extract its coefficients to obtain a closed form for the nth term.
• What Is a Closed-Form Summation? A Worked Example
  Learn what a closed-form summation means and how to derive and verify one, using the sum of the first n positive integers as a worked example.
• How to Calculate the Partial Volume of a Capsule Tank
  For a horizontal capsule tank with radius r and straight length L, calculate liquid volume at depth h using circular-segment and spherical-cap formulas.