Question

Convergent or divergent:
SUM 3^-n/2
If it is convergent calculate the sum

Answer 1

its convergent!

Answer 2

it sure is

Answer 3

its a geometric progression!!

Answer 4

\[\sum_{n=0}^{\infty} \]\[3^{-n}/2\]

Answer 5

could you please explain why it it convergent?

Answer 6

cause if we start with n =0 ,1,2 you see that the terms keep getting smaller and smaller.. so its convergent.. only if the terms either remain the same value or keep getting bigger, its a divergent!

Answer 7

ahhh i get it, but what about the sum?

Answer 8

@Mashy that's an awful argument.
\[\sum_{n=1}^{\infty} \frac{1}{n}\]
also has decreasing terms but is not convergent.

Answer 9

However is true that the sum of a geometric series--- a, ar, ar^2, ar^3, ...---is convergent if and only if |r| < 1. In this case r = 1/3, so the sum does indeed converge.

Answer 10

r= 1/3?!

Answer 11

how can i sole 3^-n / 2 into a standard geometric formula?

Answer 12

james.. 1/n for n = 1 to infinity is A CONVERGENT

Answer 13

At the sum converges to a/(1-r)

Answer 14

@Mashy: bet you $100. Want to take the bet?

Answer 15

lol that would be an absurd bet.. why don't you enlighten me in that case? whats the answer for that???

Answer 16

i know the formula of a/(1-r)

but is a 1/2 and what is r?

Answer 17

r = 1/3

Answer 18

i have problems with the \[3^{-n}\]

Answer 19

Just write down the first few term, call them the a_n: a1 = 1/6, a2 = 1/18, a3 = 1/72 ....

Answer 20

@Mashy: We bound below the sum of the series by the sum of another series and show that that lower bound diverges:

1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + (1/9 + 1/10 + ... + 1/16) +...
>1 + 1/2 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + (1/16 + .... + 1/16) + ...
= 1 + 1/2 + 1/2 + 1/2 + 1/2 + ....

It's clear we can write down as many 1/2 terms as we want. Hence this lower sum diverges to +infinity and hence the original sum does as well.

Answer 21

So it's clear that
a1 = a
a2 = ar
a3 = ar^2

where a = 1/6 and r = 1/3 = 3^(-1)

Answer 22

wow james... i take back.. glad i didn't bet lol.. i learnt something today :D thank you!!!!

Answer 23

I wish you had. $100 would have come in handy. Next time.

Answer 24

;-)