Question

interval of convergence of power series

Answer 1

help how to i test the intervals

Answer 2

your final job is to check at \(x=4\)and \(x=-4\)right?

Answer 3

actually you should have written
\[|\frac{x}{4}|<1\]so
\[|x|<4\] making it
\[-4<x<4\] and now you need to check at the endpoints to see if the interval is open or close or whatever

Answer 4

oh nvm i see you have that somewhere on your paper

Answer 5

@satellite73 do i plug it to check do i only check x=-4,x=4 or also 0? cause i think both -4 and 4 diverge but im not sure if its correct

Answer 6

i doubt it will converge at the endpoints
one will give you
\[\sum(-1)^{n+1}\] and the other will give
\[\sum1\]

Answer 7

of course it converges at zero, it is in the interval

Answer 8

on the other hand it is identically zero at 0 so yes it converges there

Answer 9

@satellite73 okay so it diverges at both points right (-4 and 4)

Answer 10

yes i believe so

Answer 11

if you replace \(x\) by \(4\) you get
\[\sum\frac{(-1)^{n+1}\times 4^n}{4^n}\]
\[=\sum(-1)^{n+1}\]which does not converge

Answer 12

if you replace \(x\) by \(-4\)you either get
\[\sum1\] or
\[\sum-1\] not sure, but it diverges as well

Answer 13

@satellite73 okay yea since it goes to 1 or -1 is not consistent so i think it diverges as well thank you

Answer 14

yw