Power Series $0^{0}$

Question

My textbook explains that the power series: $\sum_{n=0}^{\infty} x^{n}/n!$ converges for $x=0$ because the terms of the series get the value 0.

My problem with this argument is the first term, which is $0^{0}$. But this is undefined? Someone who can explain this?

The first term is taken to be $1$ for all $x$. A less compact way of writing the series is $$1 + \sum_{n=1}^{\infty}\frac{x^n}{n!},$$ since it is the Taylor series of $e^x$, so the constant term is $f(0) = e^0 = 1$. — Arturo Magidin, May 27 '12 at 22:07
This convention is nearly universally observed to prevent the awkwardness Arturo has described. — ncmathsadist, May 27 '12 at 22:11
People sometimes believe that $0^0$ can't be defined because they feel that a discontinuity is the same thing as a contradiction. In case you're wondering in which contexts you can replace $0^0$ by $1$, there is an easy to remember rule for that: It's always OK. — Mark, Feb 16 '17 at 22:35

score 6 · Accepted Answer · edited May 27 '12 at 23:31

6

In the context described in the question, it is a convention that $0^0 = 1$.

edited May 27 '12 at 23:31

answered May 27 '12 at 22:12

davidlowryduda

94,345

score 1 · Answer 2 · answered May 28 '12 at 01:10

In a power series, $x^n$ is not exponentiation operation on real numbers. Instead, it uses a different exponentiation operation; e.g. repeated multiplication suffices for this purpose. Therefore, $x^0=1$ identically.

Of course, most people don't like to pay attention to this level of detail in syntax, so they just treat $x^0=1$ when $x=0$ as a convention.

Mark · Answer 3 · 2017-03-04T18:45:15.917

0

There are calculus books that say that $0^0$ is undefined. The reason for this is tradition; long ago, before continuous functions were well understood, Cauchy placed $0^0$ in a table of "indeterminate forms", a concept that becomes obsolete once you know the relation between limits and continuous functions.

There are numerous places in mathematics where $0^0$ is implicitly assumed to be 1. So if you want consistency, then $0^0$ must be defined as 1.

Some people say that sometimes 0 is better and sometimes 1 is better, but this is not true, the value 0 is never useful, and the value 1 never leads to contradictions.

edited Mar 04 '17 at 18:45

answered Apr 17 '14 at 18:38

Mark

1,305

1

Welcome to MSE. I feel compelled to point out that your usage of "never" seems at odds with the customary mathematical definition. How does $$\lim_{x\to 0} (e^{-k/x^{2}})^{x^{2}} = e^{-k}, k \geq 0 \text{ arbitrary},$$fit into your linguistic scheme? – Andrew D. Hwang Apr 17 '14 at 19:05
Setting $0^0 = 1$ never leads to contradictions if you follow the rules of mathematics (if you believe that a discontinuity is the same thing as a contradiction then all bets are off). – Mark Feb 16 '17 at 22:31

Power Series $0^{0}$

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