Question

.

Answer 1

Ratio test

Answer 2

Which said it didn't work on it problem, because it says the series diverges? But I don't think it's supposed to diverge

Answer 3

you testing

lim abs ( a_(n+1) / a_n) ) = L
n->inf

if L < 0, the series converge
if L > 0, the series diverge
if L = 1, it's inconclusive

did you try that?

Answer 4

Yeah, and it came out as diverging for some reason

Answer 5

you're only testing {[((-1)^(n))*((x^2)^(2n))]/(((2n)!))}

not {[((-1)^(n))*((x^2)^(2n))]/(((2n)!))} - 1.

-1 is not part of the sum

Answer 6

Why is -1 not part of it?

Answer 7

It is, I just may not have written it that way in the question, but it is part of the sum in the question.

Answer 8

that is because {[((-1)^(n))*((x^2)^(2n))]/(((2n)!))} is the series for cos(x^2) only. Not cos(x^2) - 1

it's like this
-1 + cos(x^2)
-1 + sum{0->inf} {[((-1)^(n))*((x^2)^(2n))]/(((2n)!))},

Answer 9

Oh, I see. So would the radius of convergence still be (-inf, +inf) then?

Answer 10

yes

Answer 11

What makes it be (-inf, +inf)? Is it proven by the ratio test or just because it converges or what?

Answer 12

by ratio test

Answer 13

okay, thank you