Question

conditionally convergent?

Answer 1

\[\sum_{n=0}^{\infty}(x-2)^n/10^n\]

Answer 2

yes it is conditionally convergent. Earlier we found that its interval of convergence was
-8 < x < 12

If x isnt within those boundaries, it diverges.

Answer 3

book says no

Answer 4

it says for what values of x it is conditionally convergent: none

Answer 5

i wonder how it can be no =/ its basically in the form:
\[\sum_{n=0}^{\infty}r^n\]
itsa geometric series.

Answer 6

Oh, for what values of x. That is correct. There is no value of x that is conditionally convergent.

Answer 7

i was thinking the series as a whole. my bad >.<

Answer 8

we think in the interval that we found?

Answer 9

i cant understand

Answer 10

im a little lost as to what their definition of "conditionally convergent" is

Answer 11

just we look absolute value of the series

Answer 12

ah, i see. So we need to check if:
\[\sum_{n=0}^{\infty}\left| \left(\frac{x-2}{10}\right)^n \right|\]
is convergent.

Answer 13

Which it is. The geometric series is convergent as long as |r| < 1, it doesnt matter if its positive of negative.

Answer 14

i said wrong for the conditionally convergent sorry

Answer 15

still i am confused

Answer 16

for 3 =x it converges

Answer 17

basically you have two series:
\[\sum_{n=0}^{\infty}a_n\] and
\[\sum_{n=0}^{\infty}\left| a_n \right|\]
if they both converge, then the series is absolutely convergent. If only one converges, then its conditionally convergent.

Answer 18

In our case, they both converge, so its absolutely convergent, not conditionally convergent.

Answer 19

ok got it now thanks again

Answer 20

what is the reason for not conditionally convergent , cant it be conditionally and absolutely convergent?

Answer 21

at the same time?

Answer 22

its gotta be one or the other. Either both conditions are met, or only one of them. If both, its absolute. If one, conditional.

Answer 23

if this is the rule ,ok. but still it is not relevant for me ,thanks