What is the limit of the sequence?

[arl146]

Homework Statement

the problem asks to find the limit of the sequence a_n = (1+(2/n))^n

Homework Equations

?

The Attempt at a Solution

i saw a solution where they said:

lim{n→∞} [1+(2/n)]^n
= lim{n→∞} [1+(2/n)]^[(n/2)2]
= [lim{m→∞} [1+(1/m)]^m]^2, where m = n/2
= e^2

am i supposed to just know that [1+(1/m)]^m = e ? or am i supposed to show it for this problem

 

[Office_Shredder]

am i supposed to just know that [1+(1/m)]^m = e ? or am i supposed to show it for this problem

The limit as m goes to infinity is e. Either you're supposed to know this or you're supposed to prove it... what is the definition of e that you are using?

 

[Mark44]

Your book probably has an example where they let y = (1 + 1/m)^m, then take ln of both sides.

After that they take the limit and use L'Hopital's Rule. There's a little more to it than I've said, but that should give you something to look for.

 

[arl146]

i don't know what the definition of e is that I am using ...

i just don't know in my homework if i could just get to that point and then say,oh yea [1+(1/m)]^m = e and that be it. or if i actually have to show it. i guess it doesn't hurt to show it. ill try it out

 

[Mark44]

i don't know what the definition of e is that I am using ...

i just don't know in my homework if i could just get to that point and then say,oh yea [1+(1/m)]^m = e and that be it. or if i actually have to show it. i guess it doesn't hurt to show it. ill try it out

You don't need the definition of e. Use what I said in post #3.