Question

Does the series converge or diverge?

N=1 and b=infinity of (1/2)^n+(2/3)^n

Answer 1

Well, you have the addition of two geo series really.
\[\sum_{n=1}^{\infty}(\frac{ 1 }{ 2 })^{n} + \sum_{n=1}^{\infty}(\frac{ 2 }{ 3 })^{n}\]

Answer 2

Yup:)

Answer 3

For a geo-series in the form of ar^n, if the absolute value of r, (1/2) and (2/3) would be your r's in this case, is between 0 and 1, the sequence diverges. What it diverges to is
\[\frac{ a }{ 1-r } \]But then again, if you dont need the actual sum I suppose you could stop xD

Answer 4

\[\{\left(\frac{1}{2}\right)^n+\left(\frac{2}{3}\right)^n\}_{n=1}^{\infty}\] a sequence, or a series ?

Answer 5

oh nvm it says series
what @Psymon wrote

Answer 6

except where @Psymon wrote "diverge" it should be "converge"

Answer 7

Yeah, accident xD

Answer 8

no doubt a typo
"For a geo-series in the form of ar^n, if the absolute value of r, (1/2) and (2/3) would be your r's in this case, is between 0 and 1, the sequence CONverges. What it CONerges to is "

Answer 9

guess I was being lazy about the specifics of it x_X

Answer 10

Oh thank you!!

Answer 11

Brb!

Answer 12

Lol, okay O.o

Answer 13

back:D sorry i had to get gas for my car!

Answer 14

was it expensive?

Answer 15

yeah as always:(

Answer 16

lol
did you add this stuff up yet, or are you done?

Answer 17

its actually pretty cheap though but it still sucks ya know lol

Answer 18

no i havent added anything yet

Answer 19

so do i just put 1 in for n?

Answer 20

Least ya got a car, lol.

Answer 21

no you do as before
\[\sum_{n=1}^{\infty}\frac{1}{2^n}=\frac{\frac{1}{2}}{1-\frac{1}{2}}=1\]

Answer 22

true!

Answer 23

im very thankful

Answer 24

mine likes lots of gas

Answer 25

Doesnt matter for me, I couldnt drive even if I had a car xD

Answer 26

where did u get (1/2^n)?

Answer 27

\(\left(\frac{1}{2}\right)^n=\frac{1}{2^n}\) i just wrote it differently

Answer 28

i forget whether you had to add this up, or just say if it converged or not

Answer 29

do i have to do (2/3)^n as well?

Answer 30

and idk what u mean? like add up both sides?

Answer 31

yes the same way
\[\sum_{n=1}^{\infty}\left(\frac{2}{3}\right)^n=\frac{\frac{2}{3}}{1-\frac{2}{3}}\]

Answer 32

because
\[\sum_{n=1}^{\infty}(\frac{1}{2})^n+(\frac{2}{3})^n=\sum_{n=1}^{\infty}(\frac{1}{2})^n+\sum_{n=1}^{\infty}(\frac{2}{3})^n\]so you can add them up separately

Answer 33

Just to add a small think. If the objective is just to find out if this series is convergent, we don't really need to find the sum. After we split the series as psymon did, we say that the series is the sum of two geometric series each with ratio less than 1. So each series is convergent and so does the sum.

Answer 34

@satellite73 so if u add them up its 3?

@watchmath thank you...so its converges as long as the limits aren't zero right?

Answer 35

no, we don't look at any limit here. We use the fact that a geometric series is convergent if only if the ratio is less than 1.

Answer 36

no if the limit of the sum was somehow zero, then it would converge to zero
and yes, the sum is 3

Answer 37

so the fact that the sum is 3 tells me what exactly? im sorry im just getting a little mixed up...theres so much information that goes into these problems!

Answer 38

the fact that you can actually add these up means the sum converges, it converges to that number

Answer 39

awesome:) makes sense to me!

Answer 40

but you might have a sum that you know converges
but you could not be able to find that number
you would just know it converged by some test or other, but you would not know how to actually compute the sum

Answer 41

in this case you not only know that it converges, but you know what it converges to, namely 3

Answer 42

perfect that makes sense!

Answer 43

thank you!

Answer 44

yw

Answer 45

do u mind helping me with a couple more? ill repost them

Answer 46

if i can

Answer 47

post in a new thread, you will get more responses
some may actually be right