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If I have the following series:

$$(1-x)^{-1} = \sum_{n=0}^\infty x^n $$

and I want to differentiate it to get:

$$(1-x)^{-2} = \sum_{n=0}^\infty nx^{n-1} $$

At this point, is it necessary to shift this series to get rid of the first term which is 0 and start it at 1 to get:

$$(1-x)^{-2} = \sum_{n=1}^\infty nx^{n-1} $$

If I do not shift the series and leave it starting with 0, is it considered wrong?

Thank you!

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    It avoids a technical nuisance, since if $x=0$ the formula would give rise to $0 \cdot {1 \over 0}$. – 2017-02-03
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    Ah I see what you mean. If however, for example I had a series where there wasn't a technical issue with starting the series at 0, simply that the first term would be 0, would starting it with 0 instead of 1 be wrong then? – 2017-02-03
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    If it is a real number multiplied by $n=0$ then it is fine. – 2017-02-03

1 Answers 1

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You are not shifting the series but simply expressing the well-known rule

$$1'=0$$

which you have to reason to write

$$1'=0x^{-1}.$$

This term never existed.

  • 0
    I understand that if I start the series with 0, that I will be expressing the rule you just pointed out, which just shows the first term being 0. So does that mean that starting it with 0 is not wrong? – 2017-02-03
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    The term $0x^{-1}$ is not advisable as it might be considered undefined for $x=0$. – 2017-02-03
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    I see! So in a case like this I would definitely have to start the series at 1 or else starting it at 0 would be wrong? – 2017-02-03
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    @aa21: are you going to ask the question forever ? – 2017-02-03
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    It's been a long day... Thanks for the help though, sass perhaps unneeded. – 2017-02-03