13–52. Limits of sequences
Find the limit of the followi...

Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{(1 + (2 / n))ⁿ}

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Evaluate the limit of the sequence given by C subscript n is equal to parentheses of 1 + 5 divided by n to the power of N as N approaches positive infinity or state if the sequence diverges. Awesome. So it appears for this particular problem we're asked to investigate this particular sequence that is given to us by the problem itself. So we're asked to take this sequence and we're asked to evaluate it if we can, and if we cannot evaluate it, we need to state if the sequence diverges. So now that we know that we're ultimately trying to evaluate this limit of the sequence. That is given to us by the prom itself. Let's read off this, let's read off our multiple choice answers to see what our final answer might be. So A is 1, B is E to the power of 2, C is E to the power of 5, and D is diverges. Awesome. Our first step that we need to take in order to solve this problem is we need to set up the limit. So we need to say that capital L is equal to the limit, as N approaches positive infinity of parentheses 1 + 5 divided by N to the power of N. Our second step now is we need to take the natural logarithm of both sides, so we need to take the natural log of capital L as equal to the limit, as N approaches positive infinity of the natural log of parentheses 1 + 5 divided by N to the power of N. Fantastic, which we can go ahead and write that the natural log of L, which I'll just call L to keep it simple, but note that when I say L, I'm saying capital L. So the natural log of L is equal to the limit, as N approaches positive infinity of N multiplied by the natural log of 1 + 5 divided by N. Fantastic. So now our third step is we need to rewrite the expression for easier evaluation. So let us state that X is equal to 1 divided by N. So as N approaches positive infinity, let us note that X will approach 0 from the right. Thus, N will be equal to 1 divided by X. OK. So, with that in mind, we can state that N multiplied by the natural log of 1 + 5 divided by N will be equal to 1 divided by X multiplied by the natural log of 1 + 5 X. Fantastic. So our 4th step now is we need to take the limit as X approaches 0 from the right. Fantastic. So when you take the natural log of L as equal to the limit, as N approaches positive infinity of N multiplied by the natural log of 1 + 5 divided by N, which is equal to the limit, as X approaches 0 from the right of the natural log of 1 + 5 X divided by X. Fantastic. And we need to know at this stage, we need to be able to recognize that this is an indeterminate form of 0 divided by 0, and because we get an indeterminate form, we must recall and apply Lajapatal's rule. So with that in mind, our 5th step is we once again we're recalling and applying Lajapatal's rule, so we need to differentiate the numerator and the denominator with respect to X. So when you take the limit as X approaches 0 from the right of D by DX. Of the natural log of 1 + 5 multiplied by X. All divided by. D by DX of X. Which will be equal to drumroll, which will be equal to the limit, as X approaches 0 from the right of 5 divided by 1 + 5 X divided by 1. Which will be equal to the limit, as X approaches 0 from the right of 5 divided by 1 + 5 X, which will be simply equal to 5. So now our 6th and final step is we need to substitute back into our initial equation to solve for the value of capital L. So the natural law of capital L is equal to 5, so that will mean that when we rearrange this expression that I solve for L all by itself, we will find that L is equal to E to the power of 5, and that is our final answer. All right, we did it. So looking back at our multiple choice answers, the correct answer without a doubt will have to be multiple choice answer letter C, which once again is E to the power of 5. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.

{(1 + (2 / n))ⁿ}

Key Concepts

Limit of a Sequence

Exponential Limit Form (1 + 1/n)^n

Properties of the Number e

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