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!
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?
Yes . This is because .
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.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?