Question

Find the sum of the series. For what values of the variable does the series converge to this sum?

1+(x/2)+(x^2/4)+(x^3/8)+....

Answer 1

Looks like

\[\sum_{k=0}^{\infty}\left(\frac{x}{2}\right)^k\]

Answer 2

okay:)

Answer 3

This site is driving me crazy with the lag...oh well

you need \[\left|\frac{x}{2}\right|<1\]
in order to have convergence

Answer 4

can i use the infinite or finite equation?

Answer 5

what do you mean?

Answer 6

a/1-r or a(1-r^n/1-r)

Answer 7

@agent0smith do u know how to do this?

Answer 8

my test is in like 3 hours! its crunch time!!!

Answer 9

@Coolsector do u know how to do this?

Answer 10

so it is a geometric series

sum = 1/(1-q)

Answer 11

where q in our case is x/2

Answer 12

and it will converge only if |q|<1

Answer 13

how do u know that its a geometric series? u just subtract the third term from the second and the second from the first term to see if they are the same....which is also "r" correct?

Answer 14

well q in ur case

Answer 15

^ divide, remember Megan, subtracting is for an arithmetic series. and if you divide, yes it should be geometric. Geometrics only converge is r < 1.

Answer 16

and how do u know to use that formula and not a((1-r^n)/(1-r))?

Answer 17

yes you see that every term is multiplied by x/2

Answer 18

a((1-r^n)/(1-r)) - > this is for the finite case

Answer 19

^ that formula only works if a geometric series converges.

Answer 20

^and that too.

Answer 21

btw this question wasn't asking you to find the sum, that's what that formula and the infinite formula are for. It's only asking you to find what values of x is it even possible to get an infinite sum.

Answer 22

how can i tell the difference between a finite and infinite series?

Answer 23

the "..." at the end without specific term means that it is infinite

Answer 24

gotta do some tests

Answer 25

nvm i thought u mean for convergence or diveregence

Answer 26

if otherwise it was
1+(x/2)+(x^2/4)+(x^3/8)+....+ (x^8/2^8)
you would know that is finite

Answer 27

beautiful thats exactly what i needed to know...thank you!

Answer 28

yw

Answer 29

so since i am finding the sum of this series and it has ..... at the end its infinite?

Answer 30

infinite series yes

Answer 31

so i use the formula a/1-r?

Answer 32

yes and the condition for the sum to converge is |r| < 1

Answer 33

don't forget you still need to solve this for x (since this is your r) \[\large \left|\frac{x}{2}\right|<1\]since that'll give the values it converges for.

Answer 34

beautiful

Answer 35

how is x/2 my r?
and why do i have to solve for x? @agent0smith

Answer 36

wouldnt it be x^n-x/2?

Answer 37

(x/2) is your r since if you divide one term by the one before it, or just notice that each term is multiplied by x/2 to get the next one - that is r.

Answer 38

This is why you have to solve for x "For what values of the variable does the series converge to this sum?"

Answer 39

okay i got it thank you:)))

Answer 40

k :)