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I know that this is a very basic well known concept but recently I stumbled across a video on Youtube in which the creator argues that the two are not equivalent

I posted a comment arguing that in the case of Infinite sum of Σ

_{n=0}9(1/10)

^{n}you can find the sum of the series using the equation a/(1-r) which gives 9/(1-(1/10)) which equals 10

Thus the sequence 9 + 0.9 + 0.09... which should equal 9.999... also equals 10 and subtracting 9 from both values gives 0.999... equals 1

The creator of the video responded to this by stating that the partial sum of the series approaches the value of 1 but never reaches it which I suppose is true for limits however the fact that it is an infinite sum should lend it to be equal to what it approaches should it not?

This seems to simply be an argument about terminology and I am by no means a mathematics expert so I was curious as the what some of you guys had to say in regard to this