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Is there a difference in the meaning of Taylor Series and Taylor Series Expansion? For example:

The Taylor Series of the exponential function about $0$ is: $$e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}$$

Whereas the Taylor Series Expansion about $0$ is: $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$$

Is this correct? I don't mean to be overly pedantic, but I am interested in knowing if there is actually a difference in their definitions. Thank you!

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    $\begingroup$ Difference between what and what? In fact $\sum\frac{x^n}{x!}$ and $1+x+\dots$ are exactly the same expansion, in different notation. So you asking about the difference between something and itself. $\endgroup$
    – David C. Ullrich
    Commented Apr 30, 2020 at 13:15
  • $\begingroup$ I am asking if there is a difference. Perhaps Taylor Series is strictly reserved for the summation notation where as Taylor Series Expansion is reserved for the "expanded" notation. I know that both are equal to $e^x$, I was just wondering about the wording. $\endgroup$
    – Crystal
    Commented Apr 30, 2020 at 14:13
  • $\begingroup$ "I am asking if there is a difference." I understood that - that's why I explained that there is no difference. There is no difference between $4$ and $2+2$; they're exactly the same thing $\endgroup$
    – David C. Ullrich
    Commented Apr 30, 2020 at 22:23

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They are the same. The Taylor series is an expansion of a function into an infinite sum.

Both the Sigma notation and the pattern form you listed are equivalent—the summation form is somewhat more rigorous when it comes to notation as it provides a precise definition for the infinite series, while the second way you wrote is a way to quickly grasp intuitively what the series looks like. But it has nothing to do with one being an expansion and the other not.

The goal is for a person reading to be able to grasp precisely how one constructs the terms in the infinite series, so authors will use the representations interchangeably depending on the context.

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  • $\begingroup$ Thank you very much, that clears everything up! $\endgroup$
    – Crystal
    Commented Apr 30, 2020 at 14:14

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