# Approximating functions with Polynomials from Taylor Series

I wish to approximate the value of $$e$$ and show this is accurate to at least the sixth digit. I am wondering if in doing so I need to use the Taylor Series for $$e^x$$ about $$1$$ or does it suffice to use $$\Sigma_{n=0}^{\infty}\frac{x^n}{n!}$$ with $$x=1$$.

From here I believe I then would need to solve $$|R_n(x)|<|\frac{f^{n+1}(c)}{(n+1)!}|=|\frac{3}{(n+1)!}|<.0000005$$ which would work for $$n=10$$. Then just calculate $$P_{10}(x)$$. I have already shown that $$e^x<3$$ which is how I bounded the remainder function.

Does this work or should I redo the process with the Taylor Series centered about the value which we wish to approximate?

• You just showed that the remainder, i.e. the difference between the computed value and the true value is within tolerance. What more do you want ?
– user65203
Dec 12 '19 at 16:26
• I was just wondering if I should have used the Taylor Series centered around $x=1$ to estimate it at that value and a reasoning for why Dec 12 '19 at 16:27
• What is the rationale of recomputing the value when you have proven that you have the right one ?
– user65203
Dec 12 '19 at 16:31

This works. It was possible, because you know the values of $$e^x$$ and its derivatives, at 0 (they all equal 1). If you centre the series at 1 instead, then you need to know the value of $$e^x$$ at $$x=1$$, which is the very thing that you hope to approximate !
• ah perfect thank you so much! So in general, can we approximate $e^x$ with the Taylor Series at $x=0$ for any value of $x$ within the interval of convergence Dec 12 '19 at 16:28