Question

find the sum of each of the followinh series or state that the series is divergent.
a) 1+1/3+1/9+1/27+...
b) 1- 2/3 + 4/9 - 8/27 +...
c) 1/4 - 5/16 +25/64 - 125/256 +...

Answer 1

I'm not sure how to approach them and getting the common ratio... please help...

Answer 2

start by finding the common ratio

Answer 3

common ratio = `next term` / `present term`

Answer 4

a) 1+1/3+1/9+1/27+...

if `present term` = 1, then `next term` would be 1/3

common ratio = `1/3` / `1` = 1/3

Answer 5

oh... i didn't know that you divide them to get the common ratio.

Answer 6

Yes, see if you can find the common ratio for second series :
`b) 1- 2/3 + 4/9 - 8/27 +...`

Answer 7

i'm not sure because of the plus sign and minus sign, but i got 4/9 as the answer

Answer 8

simply take the ratio :
\[r = \dfrac{-2/3}{1} = -2/3\]

Answer 9

oh... i thought i need to subtract and then divide...

Answer 10

keep in mind, for the common ratio you just take the ratio of `next term` and `present term`

Answer 11

try to find the common ratio for third sequence :
`c) 1/4 - 5/16 +25/64 - 125/256 +...`

Answer 12

is it 5/4?

Answer 13

thats almost correct! it should be -5/4

Answer 14

Now that we have the common ratios, lets find the infinite sum if they exist...

do you know the infinite sum formula ?

Answer 15

yes. it's S=a/1-r

Answer 16

yes thats the one!

Answer 17

Not all infinite sums converge, do you know the criterian for an inifnite geometric sum to converge ?

Answer 18

no, can you please explain... thanks!

Answer 19

Easy :
the infinite geometric series converges only if the common ratio is between -1 and 1

Answer 20

why?

Answer 21

thats a good question, we can work that out by taking the limit of partial sum formul

Answer 22

see if this helps in deriving the criterian for convergence
https://www.khanacademy.org/math/precalculus/seq_induction/infinite-geometric-series/v/infinite-geometric-series

Answer 23

After watching that video try to figure out which of the given series converge

Answer 24

just apply the infinite sum formula for the series that converge

Answer 25

i think i know... a) and b) are convergent and c) is divergent!