Taylor and Maclaurin (Power) Series Calculator

The calculator will find the Taylor (or power) series expansion of the given function around the given point, with steps shown. You can specify the order of the Taylor polynomial. If you want the Maclaurin polynomial, just set the point to `0`.

Show Instructions
  • In general, you can skip the multiplication sign, so `5x` is equivalent to `5*x`.
  • In general, you can skip parentheses, but be very careful: e^3x is `e^3x`, and e^(3x) is `e^(3x)`.
  • Also, be careful when you write fractions: 1/x^2 ln(x) is `1/x^2 ln(x)`, and 1/(x^2 ln(x)) is `1/(x^2 ln(x))`.
  • If you skip parentheses or a multiplication sign, type at least a whitespace, i.e. write sin x (or even better sin(x)) instead of sinx.
  • Sometimes I see expressions like tan^2xsec^3x: this will be parsed as `tan^(2*3)(x sec(x))`. To get `tan^2(x)sec^3(x)`, use parentheses: tan^2(x)sec^3(x).
  • Similarly, tanxsec^3x will be parsed as `tan(xsec^3(x))`. To get `tan(x)sec^3(x)`, use parentheses: tan(x)sec^3(x).
  • From the table below, you can notice that sech is not supported, but you can still enter it using the identity `sech(x)=1/cosh(x)`.
  • If you get an error, double check your expression, add parentheses and multiplication signs where needed, and consult the table below.
  • All suggestions and improvements are welcome. Please leave them in comments.
The following table contains the supported operations and functions:
TypeGet
Constants
ee
pi`pi`
ii (imaginary unit)
Operations
a+ba+b
a-ba-b
a*b`a*b`
a^b, a**b`a^b`
sqrt(x), x^(1/2)`sqrt(x)`
cbrt(x), x^(1/3)`root(3)(x)`
root(x,n), x^(1/n)`root(n)(x)`
x^(a/b)`x^(a/b)`
abs(x)`|x|`
Functions
e^x`e^x`
ln(x), log(x)ln(x)
ln(x)/ln(a)`log_a(x)`
Trigonometric Functions
sin(x)sin(x)
cos(x)cos(x)
tan(x)tan(x), tg(x)
cot(x)cot(x), ctg(x)
sec(x)sec(x)
csc(x)csc(x), cosec(x)
Inverse Trigonometric Functions
asin(x), arcsin(x), sin^-1(x)asin(x)
acos(x), arccos(x), cos^-1(x)acos(x)
atan(x), arctan(x), tan^-1(x)atan(x)
acot(x), arccot(x), cot^-1(x)acot(x)
asec(x), arcsec(x), sec^-1(x)asec(x)
acsc(x), arccsc(x), csc^-1(x)acsc(x)
Hyperbolic Functions
sinh(x)sinh(x)
cosh(x)cosh(x)
tanh(x)tanh(x)
coth(x)coth(x)
1/cosh(x)sech(x)
1/sinh(x)csch(x)
Inverse Hyperbolic Functions
asinh(x), arcsinh(x), sinh^-1(x)asinh(x)
acosh(x), arccosh(x), cosh^-1(x)acosh(x)
atanh(x), arctanh(x), tanh^-1(x)atanh(x)
acoth(x), arccoth(x), cot^-1(x)acoth(x)
acosh(1/x)asech(x)
asinh(1/x)acsch(x)

Enter a function:

Enter a point:

for Maclaurin series, set the point to `0`

Order `n=`

Write all suggestions in comments below.

Solution

Your input: calculate the Taylor (Maclaurin) series of $$$\sin{\left (x \right )}$$$ up to $$$n=5$$$.

A Maclaurin series is given by $$$f\left(x\right)=\sum\limits_{k=0}^{\infty}\frac{f^{(k)}\left(a\right)}{k!}x^k$$$.

In our case, $$$f\left(x\right)\approx\sum\limits_{k=0}^{n}\frac{f^{(k)}\left(a\right)}{k!}x^k=\sum\limits_{k=0}^{5}\frac{f^{(k)}\left(a\right)}{k!}x^k$$$.

So, what we need to do to get the desired polynomial is to calculate the derivatives, evaluate them at the given point, and plug the results into the given formula.

$$$f^{(0)}\left(x\right)=f\left(x\right)=\sin{\left (x \right )}$$$.

Evaluate the function at the point: $$$f\left(0\right)=0$$$.

  1. Find the 1st derivative: $$$f^{(1)}\left(x\right)=\left(f^{(0)}\left(x\right)\right)^{\prime}=\left(\sin{\left (x \right )}\right)^{\prime}=\cos{\left (x \right )}$$$ (steps can be seen here).

    Evaluate the 1st derivative at the given point: $$$\left(f\left(0\right)\right)^{\prime }=1$$$.

  2. Find the 2nd derivative: $$$f^{(2)}\left(x\right)=\left(f^{(1)}\left(x\right)\right)^{\prime}=\left(\cos{\left (x \right )}\right)^{\prime}=- \sin{\left (x \right )}$$$ (steps can be seen here).

    Evaluate the 2nd derivative at the given point: $$$\left(f\left(0\right)\right)^{\prime \prime }=0$$$.

  3. Find the 3rd derivative: $$$f^{(3)}\left(x\right)=\left(f^{(2)}\left(x\right)\right)^{\prime}=\left(- \sin{\left (x \right )}\right)^{\prime}=- \cos{\left (x \right )}$$$ (steps can be seen here).

    Evaluate the 3rd derivative at the given point: $$$\left(f\left(0\right)\right)^{\prime \prime \prime }=-1$$$.

  4. Find the 4th derivative: $$$f^{(4)}\left(x\right)=\left(f^{(3)}\left(x\right)\right)^{\prime}=\left(- \cos{\left (x \right )}\right)^{\prime}=\sin{\left (x \right )}$$$ (steps can be seen here).

    Evaluate the 4th derivative at the given point: $$$\left(f\left(0\right)\right)^{\prime \prime \prime \prime }=0$$$.

  5. Find the 5th derivative: $$$f^{(5)}\left(x\right)=\left(f^{(4)}\left(x\right)\right)^{\prime}=\left(\sin{\left (x \right )}\right)^{\prime}=\cos{\left (x \right )}$$$ (steps can be seen here).

    Evaluate the 5th derivative at the given point: $$$\left(f\left(0\right)\right)^{\left(5\right)}=1$$$.

Now, use the calculated values to get a polynomial:

$$$f\left(x\right)\approx\frac{0}{0!}x^{0}+\frac{1}{1!}x^{1}+\frac{0}{2!}x^{2}+\frac{-1}{3!}x^{3}+\frac{0}{4!}x^{4}+\frac{1}{5!}x^{5}$$$.

Finally, after simplifying we get the final answer:

$$$f\left(x\right)\approx x- \frac{1}{6}x^{3}+\frac{1}{120}x^{5}$$$.

Answer: the Taylor (Maclaurin) series of $$$\sin{\left (x \right )}$$$ up to $$$n=5$$$ is $$$\sin{\left (x \right )}\approx x- \frac{1}{6}x^{3}+\frac{1}{120}x^{5}$$$.