Limit of Infinite Series | Solving y = (2n+3n)/4n | Homework Statement

[steve2212]

Homework Statement

Find the limit of the sum of:

y = (2ⁿ + 3ⁿ) / 4ⁿ

The Attempt at a Solution

as n-> infinity, y approaches 0. I don't know where to proceed from here.

 

[Dick]

It's 2^n/4^n+3^n/4^n=(2/4)^n+(3/4)^n. It's two separate geometric series. Can you deal with those?

 

[steve2212]

It's 2^n/4^n+3^n/4^n=(2/4)^n+(3/4)^n. It's two separate geometric series. Can you deal with those?

Ya I got that but I don't know how to find the sum of an infinite geometric series

 

[Dick]

Ya I got that but I don't know how to find the sum of an infinite geometric series

That's regrettable. Then you should probably try to look it up in your text or online. It's pretty basic.

 

[steve2212]

Oh sorry I'm stupid, I remembered how haha.

Since you're here I also have another hard one.

A circle is inscribed in a triangle, a square is inscribed in that circle, a circle is inscribed is that square, a pentagon is inscribed in that circle, the trend continues with the degree going up.

How do I proceed to solve this quesiton? I need to find t he sum of the limit

 

[Dick]

Oh sorry I'm stupid, I remembered how haha.

Since you're here I also have another hard one.

A circle is inscribed in a triangle, a square is inscribed in that circle, a circle is inscribed is that square, a pentagon is inscribed in that circle, the trend continues with the degree going up.

How do I proceed to solve this quesiton? I need to find t he sum of the limit

The first one wasn't hard. That one probably is. What are to trying to find the sum or the limit of? The areas, or the perimeters or the radii, or what? Not that I know the answer. But your question isn't even clear.

 

[steve2212]

The limit of the sum of the area. Sorry forgot that detail.

 

[Dick]

The limit of the sum of the area. Sorry forgot that detail.

The sum of the areas of ALL of the geometric figures? I think it's pretty likely it diverges. Mostly gut feeling. Is this for a class, or is this your own creation? It's way out of scale with the difficulty of your first problem.

 

[steve2212]

It was a bonus question on our test yesterday. Yup we need the limit of the sum of all of the areas. I know that the areas of the figures approach to 0 that's easy.

 

[Dick]

How do you know the areas approach zero? Sure, they decrease. But that doesn't convince me that they approach zero. Do you know this has a simple solution? Because I'm sure not seeing it.

 

[steve2212]

How do you know the areas approach zero? Sure, they decrease. But that doesn't convince me that they approach zero. Do you know this has a simple solution? Because I'm sure not seeing it.

Area can't be negative, and area decreases, it has to approach 0. Right?

 

[Dick]

Area can't be negative, and area decreases, it has to approach 0. Right?

Nope. 1+1/n is a positive decreasing sequence but it doesn't approach 0. It approaches 1.

 

[steve2212]

Sorry where do you get 1+ 1/n?

 

[Dick]

Sorry where do you get 1+ 1/n?

It's just an example of a sequence that decreases but doesn't approach zero.