When do you cancel the terms of when using the ratio test for the power series?

[proud flame]

I am just a bit confused on here because in this youtube video (x-3^n) was cancelled but the n value is not even though they could be cancelled.

[proud flame]

Im assuming an n value always has to be there in order for the power series to work

[cursive trail]

Which n value can be cancelled?

[proud flame]

the n+1 and n. can it not?

u could cancel the n's

[cursive trail]

Nah you can’t. Lets say we hace just n/(n+1). Your claim is we can cancel the n’s and get 1/1=1, but if we go back to the original expression no matter what we plug in for n we will never get 1

You can only cancel multiplication factors

[proud flame]

so like x-3^n+1 can be cancelled in the numerator but not n + 1 in the denominator bc its only added

[proud flame]

no you can not

think of $a = (x-3)^n$

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[proud flame]

you have a in the denominator and a in the numerator so you can cancel them out

so if let's say $a = (x-3)^n, then the expression will become \frac{a^{n+1} n}{a^n (n+1)}$

and so you can split the expression into $\frac{a^{n+1}}{a^n}$ and $\frac{n}{n+1}$

so $a^{n+1 - n}$

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[proud flame]

and it'll be just a

so the whole thing will be $a (\frac{n}{n+1})$

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[proud flame]

we know that as n approaches infinity $\frac{n}{n+1}$ will be 1

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[proud flame]

so you're left with just a

which is $(x-3)^n$ in this case

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[reef yew]

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