Is there a way to collapse a Maclaurin expansion to the original function?

Answer by Allen Zhang:

Plug it into your calculator and guess and check!

Just kidding. But no. Not every Maclaurin. Here's why:

As you all know, the Maclaurin series formula is:

Why is it like this? Well, the series approximates the function, and grows closer to an exact approximation every term. For example, look at this picture of the function f(x) = sin(x). (In black)

The graph of the first order Maclaurin, f(x) = x is in red.

The graph of the third order Maclaurin, f(x) = x – x^3 / (3!) is in orange.

And so on.

As you can see, the series gives a relatively close graph to the function, and if we continue on to infinity, it will give an exact graph of the function.

But the point is, it is an approximation.

It is simply a tool to help calculators or us calculate the relative values of strange functions. Sure, you could go on to infinity and perform tricky maneuvers with the series below, but that's already taught in basic Calculus BC classes.

You're missing the point of Maclaurin series if you want to undo them. Might as well plug numbers into your calculator and guess and check a function.

Excellent question!

Edit: Maclaurin series have other functions, such as solving differential equations.

They (differential equations) often result in non-closed MacLaurin series, and resolving them into a closed function would be good to be able to condense the answer more.

Thank you, Carl-Fredrik Brodda, for pointing this out.

Is there a way to collapse a Maclaurin expansion to the original function?

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