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The Taylor Series Calculator computes a polynomial approximation of a function centered at any point $$a$$. A Taylor series expands a smooth function $$f(x)$$ as an infinite sum of terms derived from the function's derivatives at a single point:
$$f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!}(x - a)^k = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots$$
This representation is one of the most powerful tools in analysis, enabling us to replace complicated transcendental functions with polynomials that are easy to evaluate, differentiate, and integrate.
Taylor series serve as the mathematical backbone for numerical computing. Every time a computer evaluates $$\sin(x)$$, $$e^x$$, or $$\ln(x)$$, it is internally using a polynomial approximation derived from Taylor's theorem. The series provides a systematic way to approximate any sufficiently smooth function to arbitrary precision by including more terms.
When we truncate the Taylor series at $$n$$ terms, the approximation error is governed by Taylor's remainder theorem. The Lagrange form of the remainder states:
$$R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$$
for some $$c$$ between $$a$$ and $$x$$. This tells us the error decreases as we add more terms, provided the series converges. The factorial growth in the denominator typically ensures rapid convergence near the center point.
Several functions have especially elegant Taylor series that appear throughout mathematics and physics:
Beyond numerical computation, Taylor series are essential in physics for linearization and perturbation theory, in engineering for control system analysis, in economics for utility function approximation, and in machine learning for optimization algorithms that rely on second-order Taylor expansions (Newton's method, Hessian approximations).
Select a function, specify the center point $$a$$ and evaluation point $$x$$, then choose the number of terms. The calculator computes derivatives at $$a$$, builds the Taylor polynomial, and compares the approximation to the exact function value.
The approximation shows how closely a truncated Taylor polynomial matches the true function value. A smaller absolute error indicates better convergence. Increasing the number of terms or evaluating closer to the center point $$a$$ generally improves accuracy.
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Using terms x − x³/6 + x⁵/120 gives an excellent approximation. With 4 terms the error is less than 0.0002.
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The exponential series converges for all x, but farther from center requires more terms. Six terms gives about 98.3% accuracy at x = 2.
A Taylor series is the full infinite sum $$\sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!}(x-a)^k$$, while a Taylor polynomial truncates this at $$n$$ terms, giving a finite approximation $$P_n(x)$$. This calculator computes Taylor polynomials with up to 6 terms.
The center point $$a$$ is where all derivatives are evaluated. The approximation is most accurate near $$x = a$$ and typically degrades as $$|x - a|$$ increases. Choosing $$a$$ close to your evaluation point gives better accuracy with fewer terms.
Each term contains the factor $$(x - a)^k$$. As $$|x - a|$$ grows, these powers increase rapidly, and the polynomial diverges from the true function unless enough terms are included to compensate.
A function must be infinitely differentiable (smooth) at the center point to have a Taylor series. Even then, the series may not converge to the function everywhere. For example, $$e^{-1/x^2}$$ is smooth but its Taylor series at $$x = 0$$ is identically zero, which does not equal the function for $$x \ne 0$$.
It depends on the function and how far $$x$$ is from $$a$$. For $$\sin(x)$$ and $$\cos(x)$$ near 0, 4–6 terms give excellent results. For the exponential at large $$x$$, you may need many more. Watch the absolute error to judge convergence.
The radius of convergence $$R$$ is the distance from $$a$$ within which the series converges. For $$e^x$$, $$\sin(x)$$, and $$\cos(x)$$, $$R = \infty$$. For $$\ln(1+x)$$ centered at 0, $$R = 1$$. For $$1/(1-x)$$ centered at 0, $$R = 1$$.
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