Question

Find the exact sum of the following series

Answer 1

\[\sum_{k=3}^{\infty} (1/2)^k\]

Answer 2

Find the partial sum and take it's limit

Answer 3

I think the answer is 1/4 since it's infinite geometric series

Answer 4

how do you find the partial sum?

Answer 5

ahh don't think that is required here unless you're taking a course in analysis

Answer 6

so how do you know that is 1/4

Answer 7

Google infinite geometric series; there's a formula for that

Answer 8

Here... you know that geometric series take this form, right?
\[\huge S=a + ar + ar^2+ar^3+...\]
right?
Where a is the first term and r is the common ratio.

Answer 9

yes

Answer 10

Well, let's rearrange.
\[\huge S-a=ar+ar^2+ar^3+ar^4+...\]following me so far? :)

Answer 11

yes

Answer 12

Okay, let's divide both sides by r, the common ratio, we get...
\[\huge \frac{S-a}{r}=a+ar+ar^2+ar^3+...\]Catch me so far?

Answer 13

yes

Answer 14

Now, you'll notice that the right-side of this equation is nothing but S again. Right?

Answer 15

yes

Answer 16

So, we now have
\[\huge \frac{S-a}{r}=S\]Now we just have to solve for S.

Answer 17

a/(1-r)

Answer 18

That's right..
\[\large S-a=Sr\]\[\large S-Sr=a\]\[\large S(1-r)=a\]\[\huge S=\frac{a}{1-r}\]

Answer 19

so whats the "a" value for this series and whats the r value

Answer 20

a is your first term
r is what you keep multiplying to the first term, over and over again.

Answer 21

so the first term is (1/2)?

Answer 22

Your series starts at k=3, though.

Answer 23

so (1/2)^3 is my first term?

Answer 24

and r = (1/2)?

Answer 25

Yep.

Answer 26

thanks for your help

Answer 27

No problem. :)