Power Series

[lament rose]

how does the top part simplify into the answer below?

[woven jackal]

1. Wait patiently for a helper to come along.
2. Once someone helps you, say thank you and close the thread with: ```diff
   +close

[wet dew]

the (-1)^n cancels out all but one of the -1 in (-1)^{n+1} which leaves the negative at the front

same with the 2^n and 2^{n+1} on top

that's it

[lament rose]

am i allowed to send photos of my work

[wet dew]

yeah

[lament rose]

okay thanks

so what ur saying

the denominator -1^n cancels the numerator -1^n+1?

[wet dew]

combine the top and get:
$\frac{(-1)^{n+1} 2^n n^2}{2^{n+1} n^{3} (-1)^n}$
cancel the... wait

[jaunty basin]

xX_69killa_agex420_Xx

[wet dew]

your workings aren't the same thing as what you posted originally

[lament rose]

the original post was from chegg

[wet dew]

ahh

got it

[lament rose]

and i thought when you solve for it originally, you would n+1

everything first

its also like this

i'm not sure how everything cancels out

[wet dew]

then you can see that you are left with an n on the bottom cuz of $n^{2-3}$

[jaunty basin]

xX_69killa_agex420_Xx

[wet dew]

and you are left with a 2 on the bottom because of $2^{n-(n+1)}$

[jaunty basin]

xX_69killa_agex420_Xx

[wet dew]

same thing with the -1^whatever

[wet dew]

with this you get your 6^n denominator cancelling out all but one of the 6s on numerator, then you have

[lament rose]

okay so the -1s cancel out

n2-n3 = 1/n

[wet dew]

$\frac{6n \ln(n)}{(n+1) \ln(n+1)}$

[jaunty basin]

xX_69killa_agex420_Xx

[wet dew]

expand denominator:
$\frac{6n \ln(n)}{n\ln(n+1)+\ln(n+1)}$

[jaunty basin]

xX_69killa_agex420_Xx

[wet dew]

and that's a fairly simple limit to solve with L'hopital 3 times if you want

[lament rose]

okay so before all the crosses

how does it equal 2^n-(n+1)

and wouldn't that just equal 2^1

[wet dew]

1/2

leaving you with $-\frac{1}{2n}$

[jaunty basin]

xX_69killa_agex420_Xx

[lament rose]

so the radius would be 2?

would it be better if i just sent the original question

[wet dew]

probably

[wet dew]

interval of convergence is simple, x≥5

yeah I think that's right for the radius

icba to do the whole question myself, sorry

[lament rose]

LOL all good

thanks for everything though

+close