Question

Hi Can someone help me show why the following sequence isn't converging when defined as followed:
x_2n-1=1/5^n , x_2n = (n)root (1/x_2n-1) .. I'll make the problem clearer in the comments.. Thanks!

Answer 1

\[x _{2n-1}=1/5^n , x _{2n} = \sqrt[n]{1/2_{n-1}}\]

Answer 2

what is the limit of (n)/(n-1)

Answer 3

1 right?

Answer 4

dunno, havent worked it out yet

Answer 5

It is 1 ..

Answer 6

When n approaches 0 its 1, it's 0 as n approaches infinity

Answer 7

\[\sqrt[n]{\frac{1}{1/5^n}}\]
\[\sqrt[n]{5^n}=5\]

Answer 8

But why is it not convergent ?

Answer 9

from the looks of it, it jumps between 0 and 5

Answer 10

so it oscillates to infinity between 0 and 5?

Answer 11

1 0.2 5
2 0.04 5
3 0.008 5
4 0.0016 5
5 0.00032 5
6 0.000064 5
7 0.0000128 5
8 0.00000256 5
9 0.000000512 5

Answer 12

So in conclusion .. That can be used to prove that this sequence does not converge ?

Answer 13

im not sure what 'proof' can be used

x_k, for odd k goes to 0
x_k for even k maintains at 5

so there is no limiting value that it settles down at

Answer 14

think of a super bouncy ball that is forever bouncing between the ceiling and the floor, where does it come to rest at?

Answer 15

Eventually 0

Answer 16

eventually it is either at the ceiling or on the floor, it never comes to rest

Answer 17

Wait sorry no it oscillates between 0 and 5 sorry lol

Answer 18

:)

Answer 19

Alright I think Iunderstand

Answer 20

Thanks!!!

Answer 21

yep ;)