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Usually when we have a sum that depends on the power of a real number, we write the null power as $1$. For example, the series

$$e^x = \sum_{k=0}^{\inf} \frac {x^k} {k!}$$

However, this doesn't work for zero, since $e^0 = 1$ but the series should be $0 + 0 +0$...

Shouldn't we, in this cases, write these sums/series by removing the $k=0$ term? Like this: $e^x = 1 + \sum_{k=1}^{\inf} \frac {x^k} {k!}$

Or do we just do a parted function a define a different formula when x=0?

Francisco José Letterio
  • 2,146
  • No we do nothing because we don't need to. Once the definition of $x^0$ has formalized 2 few centuries ago, it automatically implies that of the case for $k = 0$ can be included under the same summation sign as for the rest of the terms. – Nilotpal Sinha Nov 06 '17 at 11:10
  • $0^0 = 1$ is perfectly standard, at least in any situation where the exponent is understood to be the integer $0$, and not the real number $0$. – Arthur Nov 06 '17 at 11:22
  • For continuity of this first term, the convention $0^0$ is implicitly adopted. –  Nov 06 '17 at 11:43

1 Answers1

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You can distinguish

$$\lim_{x\to0}0^x=0$$ and $$\lim_{x\to0}x^0=1.$$

In the case of a summation, the second behavior pertains and is implicitly adopted.