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I think I have a good grasp on the fundamentals of Taylor series (what they do and how they approximate the functions), but I just don't understand how these can be useful.

For example, lets look at the following Taylor series:

$$e^x\approx 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} +\frac{x^4}{4!}+ \frac{x^5}{5!}\dotsb.$$

Why would you want to use the approximation when you have the actual equation $e^x$. It not only look simpler, but also gives you the true value of this function for any value of $x$; so why use Taylor series for which gives you just an approximation.

Maybe the point I am missing is that Taylor Series can give you an approximation on unknown functions, i.e. $f(x) = ???$.

But then you wouldn't be able to find derivatives of this function?

Could someone please help me see Taylor Series as an actual tool that can be used to solve real life problems (with an example ideally)?

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Without using a calculator, a set of tables, etc, how would find the value of $e^x$? For some people, that series actually is the definition of $e^x$.

A more general use is expanding the domain of a function e,g. from $\mathbb{R}$ to $\mathbb{C}$.

Another is integration of a function for which there is no anti-derivative.

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One simple example is the Simple pendulum, with length $l$ and in gravitational acceleration $g$. The differential equation we need to solve is: $$\frac{\mathrm{d}^2 \varphi}{\mathrm{d}t^2}+\frac{g}{l} \sin(\varphi)=0$$ Which, sadly, can't be solved analytically with the "common" functions. But if we use the first order Taylor polinomial of the $\sin$ function, i.e. $\sin(\varphi)\approx \varphi$, we get the following equation: $$\frac{\mathrm{d}^2 \varphi}{\mathrm{d}t^2}+\frac{g}{l}\varphi=0$$ Which can be solved easily: $$\varphi=\varphi_0 \cos\left(\sqrt{\frac{g}{l}}t\right)$$ Which is valid if the angle (and the ellapsed time) is small enough.

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Taylor Series might be helpful in identifying the asymptotic behavior of functions. Once we decompose a function into its Taylor Series, we sometimes see terms that are vanishing in the limit and can simplify the expression if we are only interested in its limiting behavior.

A good example of such a decomposition is the proof of Stirling's formula, where thanks to Taylor Series expansion, we identify a geometric series to complete the proof.

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There are many applications :

Computing limits

Study of continuity

Study of differentiability

Study of the sign

Finding tangent equation

Finding asymptote equation

Nature of a series

Nature of an improper integral

Nature of a singular point

Resolution of differential equations

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Polynomials are about the easiest things to deal with. They're easy to integrate and easy to differentiate. When we have non-polynomial functions, that might not be the case. Can you integrate $e^x/x$? No. But you can approximate it by a Taylor polynomial at any accuracy you like, and then easily integrate.

If you have a complicated limit, you can often replace the stubborn bits with their Taylor series and then easily find the limit.

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