Taylor Series Step by Step

You enter the function, the point at which to decompose this function, and the number of terms in the decomposition.

Decompose the function to Taylor series
Finds:
- Coefficients: $a_k$
- Power series sum: $\sum_{k=0}^{\infty} \frac{f^{(k)} (x)}{k!}\, (x - x_0)^k$
Graphing:
- The function
- Partial sums of the Taylor series
Approximate computation of the definite integral using the expansion of the integrand function into both the Maclaurin or the Taylor series (integration limits enter to the form for limits of plot)

To approximate numerical solution of the definite integral, enter:

the integration limits into additional form (see on this page)
select the calculation precision using the form field Expand into the powers of x up to the degree

Learn more about Taylor series.

The above examples also contain:

the modulus or absolute value: absolute(x) or |x|
square roots sqrt(x),
cubic roots cbrt(x)
trigonometric functions:
sinus sin(x), cosine cos(x), tangent tan(x), cotangent ctan(x)
exponential functions and exponents exp(x)
inverse trigonometric functions:
arcsine asin(x), arccosine acos(x), arctangent atan(x), arccotangent acot(x)
natural logarithms ln(x),
decimal logarithms log(x)
hyperbolic functions:
hyperbolic sine sh(x), hyperbolic cosine ch(x), hyperbolic tangent and cotangent tanh(x), ctanh(x)
inverse hyperbolic functions:
hyperbolic arcsine asinh(x), hyperbolic arccosinus acosh(x), hyperbolic arctangent atanh(x), hyperbolic arccotangent acoth(x)
other trigonometry and hyperbolic functions:
secant sec(x), cosecant csc(x), arcsecant asec(x), arccosecant acsc(x), hyperbolic secant sech(x), hyperbolic cosecant csch(x), hyperbolic arcsecant asech(x), hyperbolic arccosecant acsch(x)
rounding functions:
round down floor(x), round up ceiling(x)
the sign of a number:
sign(x)
for probability theory:
the error function erf(x) (integral of probability), Laplace function laplace(x)
Factorial of x:
x! or factorial(x)
Gamma function gamma(x)
Lambert's function LambertW(x)
Trigonometric integrals: Si(x), Ci(x), Shi(x), Chi(x)

The following operations can be performed