Question

Can someone please double check my work on this radius & interval Calc 2 question?

Answer 1

So I now have the radius .... And in order to find the interval, I need to test the two numbers by plugging them into the equation.

Answer 2

Did you test the endpoints?

Answer 3

Testing (-3):

Answer 4

(-3) converges, because it is the alternating harmonic series

Answer 5

Testing (7), (7) converges, because it is also the alternating harmonic series

Answer 6

Therefore, the Radius is 5, and the interval is [-3, 7]

Answer 7

What about the plus minus in the -3 test? Namely why did it disappear?

Answer 8

It didn't really "disappear", I just know that the alternating harmonic series converges to ln(2) or -ln(2), so the $$\pm$$ I thought would be irrelevant.

Answer 9

I'm not sure about the math behind that bit but I think this looks OK. Just wondering cause I gots me a final in a week or so

Answer 10

Me too :)

This is from a final they gave in 2011.

Answer 11

well keep bumping/tagging people and good luck with said final :D

Answer 12

Thanks! You too!

Answer 13

Anyone?

Answer 14

your results look good

Answer 15

The other way to check your radius of convergence is to use the same take the inverse of what you used to check convergence (i.e.\(\lim \cfrac{a_n}{a_{n+1}}\). In this case, you get 5, which validates your results.

Answer 16

Cool, thank you for your help!

Answer 17

yw

Answer 18

Hold on a minute, I think I found a different way to test (-3) ... doesn't this make a lot more sense?

Answer 19

@ybarrap

Answer 20

yes it does and it diverges

Answer 21

Thank you very much for the response & double check!

Answer 22

And it diverges, because the numerator will always be a positive one, since the definition of an even integer is (2n), so thus I will result with the harmonic series?

Answer 23

yes

Answer 24

-1 to any even power is just 1

Answer 25

Okay, perfect.