Many students have asked me why it is important to for to be close to zero for the Maclaurin’s expansion to be a good approximation. So here, I plot it in the 4 curves:
which is our actual curve.
which is the estimation of , up to and including .
which is the estimation of , up to and including .
which is the estimation of , up to and including .

We can observe from the graphs, that as we increase the degree of order, the estimated curves become more like that of , although it still tend to deviate a lot. The idea of maclaurin’s is that it provides us a way to interpolate and write the humble equation out in a polynomial that actually never ends. So as we continue to consider more terms, our estimation will get close and closer to the actual curve.

And clearly, we see that when , we found the actual value. This is simply because the maclaurin’s expansion is centred about zero. 🙂

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