Sequence and bounds

Question

I found that the limit of the sequence (1+22/n)^n is e^22. I'm told the sequence is bounded because the absolute value of a_n is < or = to e^22 for all positive integers. 

 

I would like to know, here and in general, how I can find if a sequence or function is bounded and then what the bound is.

 

Thank you

Answer 1

First what did you find the limit is good, correct. Since the limit exists (and finite!) the sequence is bounded (from both sides). One way to show that the given sequence is bounded is using the definition of the limit.

BUT it is desirable to show that it is bounded before you find limit.

In general, if the sequences or function might not be bounded.

As for (1+22/n)^n, it grows very fast but still limited. You can show that it is bounded from above by expanding  (1+22/n)^n as binomial expansion:

(1+22/n)^n=1+22+(n(n-1))/2!)(22/n)^2+...+n(22/n)^(n-1)+(22/n)^n<= 1+22+(n/2-1/2)*22+((n-1)(n/2-1)(n/3-1)*22.+..

or alternatively, you can use 

(1+x)^r <=e^rx.

Then (1+22/n)^n<=e^22, which basically the limit.