Taylor Series Calculator

A Taylor series is one of the most powerful ideas in calculus: it approximates a complicated function as a sum of simple power terms, built from the function's derivatives, so that even something like a sine or a logarithm can be evaluated with nothing more than additions, multiplications and divisions. This calculator brings the idea to life for the most important functions, the exponential, sine, cosine, the natural log of one plus x, and one over one minus x, using their Maclaurin series, the Taylor series centred at zero. Choose a function, set how many terms you want and the value of x, and the calculator adds up the series, shows you the approximation, compares it to the exact value, and reports the error so you can see how good the fit is. The headline lesson appears immediately: the more terms you keep, the closer the approximation gets, at least near the centre. With just a few terms the exponential and the trig functions are already accurate for small x, which is exactly why calculators and computers use these series under the hood to evaluate functions your processor cannot compute directly. It also reveals the limits: far from the centre you need many more terms, and some series, like the geometric one for one over one minus x, only converge at all within a limited range, diverging outside it. That makes the tool genuinely useful for calculus students meeting Taylor and Maclaurin series for the first time, for checking how many terms a required accuracy needs, and for building intuition about convergence. Because it recalculates as you type, you can watch the approximation tighten as you add terms or move x. The formulas and a worked example are explained clearly below.

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