Question

hope someone can help me with this- series question

Answer 1

\[\sum_{n=1}^{\infty} \frac{ 2^{n+1}}{3^{n}}\]

Answer 2

its asking for the value.

Answer 3

\[\frac{ 2^{1+1} }{ 3^1 }=\frac{4}{3}\]
Here's the pattern
\[\frac{ 4 }{ 3 }+\frac{ 8 }{ 9 }+\frac{ 16 }{ 27 }...+\frac{ 2^{n+1} }{ 3^n }\]
This is a limiting sums question, so we use this equation.
\[r=\frac{2}{3}\]
\[S _{\infty}=\frac{ a }{ 1-r }\]

Answer 4

a is the first term of this geometric sequence.

Answer 5

i understand the pattern, but how do you use the final equation? what is the A?

Answer 6

oh.

Answer 7

You should have learned all your symbols and pronumerals in the equation. If you didn't, you obviously did not pay attention in class.

Answer 8

i don't recall seeing that equation, and i was simply trying to understand the given solution for the problem.

Answer 9

Okay I will show you this equaiton. It comes from the Geometric sum equation.

Answer 10

\[S _{n}=\frac{ a(1-r^n) }{ 1-r }\]
You use this equation when:
\[\left| r \right|<1\]

Answer 11

\[r<1\]
sorry.

Answer 12

But then in your questing it states n=infinity.

Answer 13

question*

Answer 14

\[S _{\infty}=\frac{ a(1-r^\infty) }{ 1-r }\]

Answer 15

in your question r is 2/3. 2/3 is less than 1.

Answer 16

\[(\frac{ 2 }{ 3 })^{99999999}\]
When you plug in a large number such as 99999999, the denominator of the fraction becomes so large that the fraction will reach 0.

Answer 17

Because infinity is a very large number.

Answer 18

So then
\[r^{\infty}\]
becomes 0.

Answer 19

\[S _{\infty}=\frac{ a(1-0) }{ 1-r }\]
So the equation will look like this.

Answer 20

right. ok. so thats where the first equation comes from. so it should be \[\frac{ 4/3 }{ 1-2/3}\]

Answer 21

You don't need to clarify with me when you're just putting numbers where they're supposed to go, but it's correct. I won't be there to say if that's correct or not when it comes exam time.

Answer 22

i was just double checking. sorry for being sure. thanks for your help

Answer 23

No worries. Your mind should tell you it's correct. It helps your mind to be independent. You will go far if you become more independent instead of dependent. It's same case when you're crossing the road. Your parents took your hands in the early stages when you were a child and still dependent. As you get older, you become more independent and now you're crossing the road by yourself.

Answer 24

But then again, sometimes it's good to be dependent when you're in an emotional state. That's just my 2 cents for ya. Don't even need to read it.

Answer 25

haha. thanks for the mind break. its a good thought.