Thread: math homework help: infinite series

Hello!

I've been doing math homework and am going over the problems I had difficulty completing. I'm on infinite series. I'd greatly appreciate a walkthrough/explanation for the following problem and some tips on how to approach problems like this/proof problems in general.

Evaluate: **"<=" means "less than or equal to"; "An" means "Ace of n"

lim ((2^n)+(3^n))^(1/n) Hint: Show that 3<=An<=(2 x 3^n)^(1/n)
n-->infinite


This is an even problem, so I do not have a solution. I tried solving it the easy way, by which I mean setting n equal to infinite, making the equation inside to the zero power, which in turn makes the limit 1. However, because I did not use the "hint," I fear that I may have done something wrong.

Thank you!

Originally Posted by wyhwang7

Hello!

I've been doing math homework and am going over the problems I had difficulty completing. I'm on infinite series. I'd greatly appreciate a walkthrough/explanation for the following problem and some tips on how to approach problems like this/proof problems in general.

Evaluate: **"<=" means "less than or equal to"; "An" means "Ace of n"

lim ((2^n)+(3^n))^(1/n) Hint: Show that 3<=An<=(2 x 3^n)^(1/n)
n-->infinite

This is a sequence problem not a series problem!


Now use the squeeze theorem.

This problem has a simple generalization. Let be positive real numbers.
Then . Try to see if you can prove it.

Originally Posted by ThePerfectHacker

This is a sequence problem not a series problem!


Now use the squeeze theorem.

This problem has a simple generalization. Let be positive real numbers.
Then . Try to see if you can prove it.

Thank you for your help! I see how you arrived at
3 <= (2^n + 3^n)^(1/n) <= (2 x 3^n)^(1/n) by raising everything to the power of 1/n.

Using the Squeeze Theorem, would that make the limit 3?


Oh, and how did you know to use 3^n <= 2^n + 3^2 <= 3^n + 3^n

That is, how did you derive the values of Bn and Cn?

Originally Posted by wyhwang7

Thank you for your help! I see how you arrived at
3 <= (2^n + 3^n)^(1/n) <= (2 x 3^n)^(1/n) by raising everything to the power of 1/n.

Using the Squeeze Theorem, would that make the limit 3?

Yes . This is because .

Oh, and how did you know to use 3^n <= 2^n + 3^2 <= 3^n + 3^n

That is, how did you derive the values of Bn and Cn?

I just thought of what I can use to simplify the expression . And I saw that if you forget about the then the expression because which is easier to work with than . Likewise when I wrote , again it made the expression easier to work with being under power.