A guitar string never vibrates as one pure wave — it rings with a stack of overtones, and the Fourier series is the recipe that lists every one. In plain words, any repeating wave equals its average value plus a sum of sines and cosines, each vibrating at a whole-number multiple of the base frequency and each scaled by its own coefficient. Add more terms and the sum hugs the original wave more tightly; the first few terms capture the broad shape while later terms fill in the fine wiggles.

An audio engineer hunting a 60 Hz electrical hum in a studio recording reads that hum straight off the series coefficients. Power companies do the same when checking harmonics polluting the 60 Hz grid that feeds American homes.

This calculator evaluates a partial sum. Type how many terms to include into the Variable A box and the position x along the wave into the Variable B box, then read the approximated wave height from the Result box. Raise the term count and watch the approximation sharpen.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Fourier Series Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.