Limit Question using the definition of e

[RJLiberator]

Homework Statement

2*Lim (as k approaches infinity) of (| (k/(k+1))^k |)

The answer to this limit is 2/e

I know there is a definition of e used, but I am unclear what to do/how to do it. If someone has a link I can look at or could point me in the right direction I would be thankful.

 

[Dick]

Homework Statement

2*Lim (as k approaches infinity) of (| (k/(k+1))^k |)

The answer to this limit is 2/e

I know there is a definition of e used, but I am unclear what to do/how to do it. If someone has a link I can look at or could point me in the right direction I would be thankful.

The definition of e that you want is that it's the limit k->infinity (1+1/k)^k. That's pretty closely related to your limit.

 

[RJLiberator]

Ugh. I'm thinking of all the possible ways to transform this limit.

If I take a K out of every part it becomes:
(1/k)^k/(1+1/k)^k and the denominator can be replaced with e, but the numerator then goes to 0.

Am I on the right track? The limit (k-->infinity) of (1/k)^k = 0 so that can't be right.

Hmmm...

Somehow symbolab changes the equation into (k/(k+k))^k to make this work, but I have no idea how you can change 1 to k.

 

[Dick]

Ugh. I'm thinking of all the possible ways to transform this limit.

If I take a K out of every part it becomes:
(1/k)^k/(1+1/k)^k and the denominator can be replaced with e, but the numerator then goes to 0.

Am I on the right track? The limit (k-->infinity) of (1/k)^k = 0 so that can't be right.

Hmmm...

Somehow symbolab changes the equation into (k/(k+k))^k to make this work, but I have no idea how you can change 1 to k.

You are doing bad algebra. (1/k)/(1+1/k) isn't equal k/(k+1).

 

[RJLiberator]

Ah. I see.
(1/k)/(1+1/k) = 1/(k+1)

This is interesting. I feel this is getting extremely close to the goal.
However, it puzzles me. All we did was take a k out of every element of the equation to make the limit easier.
We want to correlate this to the definition of e.

If I make the definition:
1/e^k = 1/(1+1/k)^k

I seem to get closer to what my answer states, but not quite there.

 

[Dick]

Ah. I see.
(1/k)/(1+1/k) = 1/(k+1)

This is interesting. I feel this is getting extremely close to the goal.
However, it puzzles me. All we did was take a k out of every element of the equation to make the limit easier.
We want to correlate this to the definition of e.

If I make the definition:
1/e^k = 1/(1+1/k)^k

I seem to get closer to what my answer states, but not quite there.

Just fix the algebra. If you have k/(k+1) and you take a k out of numerator and denominator what do you get?

 

[RJLiberator]

Ugh... well, that was easier then I made it seem.
My algebra was off from the beginning.

Clearly, taking a K out of the numerator and denominator makes it the simple equation of
[1/(1+1/k)]^k which fits the definition of e flawlessly
1/e^k = [1/(1+1/k)]^k

Marvelous. You've been a great help here.

 

[Dick]

Ugh... well, that was easier then I made it seem.
My algebra was off from the beginning.

Clearly, taking a K out of the numerator and denominator makes it the simple equation of
[1/(1+1/k)]^k which fits the definition of e flawlessly
1/e^k = [1/(1+1/k)]^k

Marvelous. You've been a great help here.

I really hope you meant to say 1/e=limit k->infinity 1/(1+1/k)^k. That's kind of different from what you posted.

 

[RJLiberator]

I really hope you meant to say 1/e=limit k->infinity 1/(1+1/k)^k. That's kind of different from what you posted.

;)Precisely.
Thank you for pointing that out - helps me understand the definition of e, better.