Why does taylor series have ample amount of importance in calculus?

I like to know some insights behind taylor series.

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    $\begingroup$ You should distinguish between Taylor polynomials (local polynomial approximations to a finite order of a function which might only have a finite number of derivatives) and Taylor series (the limit of the sequence of Taylor polynomials of a function which has all its derivatives). These are really sufficiently different subjects that they should have separate questions. Which one do you want this to be about? $\endgroup$ – Ian Aug 22 '15 at 20:56

Taylor series use derivatives everywhere (that is one reason why they involve calculus). It is interesting to know that values of a function $f(x)$ can be approximated using only derivative information at $x=0$. For a wide collection of functions, the approximation has increasing accuracy as more derivative information is used:

$$f(x) \approx f(0) + xf’(0) + \frac{x^2}{2}f’’(0) + … + \frac{x^n}{n!} f^{(n)}(0)$$

Example (recall that $e^0=1$): \begin{align} &e^{0.1} \approx e^0 + (0.1)e^0 = 1.1\\ &e^{0.1} \approx e^0 + (0.1)e^0 + \frac{(0.1)^2}{2}e^0 = 1.105\\ &e^{0.1} \approx e^0 + (0.1)e^0 + \frac{(0.1)^2}{2}e^0 + \frac{(0.1)^3}{3!}e^0 = 1.105166666666667\\ &e^{0.1} \approx 1 + .1 + \frac{(.1)^2}{2} + \frac{(.1)^3}{3!} + \frac{(.1)^4}{4!} = 1.105170833333333 \end{align}

The approximations are converging to the true answer $e^{0.1} = 1.105170918075648…$.

In the limit (in cases when the Taylor expansion applies) we get that $f(x)$ can be exactly written as an infinite sum of polynomial functions. This is useful for understanding $f(x)$ in simpler ways. For example, differentiating or integrating $f(x)$ can be done simply by differentiating or integrating each term, which involves differentiating/integrating polynomial functions, which is easy.

Let $f(t)$ be the location of a car at time $t$. If you know where the car is at time $t=0$, and you know its velocity $v(0) = f'(0)$, then you can accurately predict where it will be at some small time later: $$ f(t) \approx f(0) + f'(0)t = f(0) + v(0)t $$ If you know both the velocity and the acceleration $a(0)=f''(0)$, then you can more accurately approximate: $$ f(t) \approx f(0) + f'(0)t + \frac{f''(0)}{2}t^2 = f(0)+v(0)t + \frac{a(0)t^2}{2} $$ You can see how this would be useful, for example, in a robotics control system that needs to guess where the robot will be in one second given its current location, velocity, and acceleration.


Polynomials are about the simplest type of functions there are. They are closed under addition, multiplication, integration, and differentiation.

Taylor series allow a function to be approximated in terms of polynomials of a specified degree. This gives a good indication of how the function behaves locally.

Note that the Taylor series of degree 1 is just the tangent.

The more terms of the Taylor series, the closer the series fits the function at the point where the series is computed.

This is a start. There is lots more.

Also look up Chebychev approximation.


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