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 + 1 / 2n)ⁿ)}

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to evaluate the limit of the sequence given by a of N, which is equal to the quantity of 1 plus 1 divided by nn quantity raises the quantity of 1 divided by n in quantity as n approaches infinity, and we're given 4 possible choices as our answers. For choice A, we have 0, for choice B, we have 1. For choice C, we have 2, and for choice D we have infinity. Now as to evaluate the limit of the sequence a ton as N approaches infinity, so we'll call that limit L. So that means that L is going to be equal to the limit as N approaches infinity of the quantity of 1 plus 1 divided by n in quantity raised to the quantity of 1 divided by n in quantity. Now, since we have an N in the exponent, it's going to be more advantageous to look at the natural log of L. So the natural log of L is going to be equal to the limit as n approaches infinity of the natural log of the quantity of 1 plus 1 divided by n in quantity raises to the quantity of 1 divided by n in quantity. To rewriting this using our properties of logarithms, this is going to be equal to the limit as n purchases infinity of 1 divided by n multiplied by the natural log of the quantity of 1 plus 1 divided by nn quantity. Now, as n approaches infinity, the quantity of 1 divided by n approaches 0. So evaluating this limit, this is going to be equal to. 0, multiplied by the natural log of the quantity of 1 + 0 in quantity, and since the natural log of 1 is 0, that means that this whole quantity is going to be equal to 0. So in order to find our value for L, we just need to exponentiate both sides of this equation. That means that L is going to be equal to E 0, which is just equal to 1. So that is the answer that we're looking for for this problem. And if we compare that to our answer choices, we see that the correct answer choice corresponds to answer choice B. So, I hope that this explanation has been useful, and I'll see you in the next video.

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

{√((1 + 1 / 2n)ⁿ)}

Key Concepts

Limits of Sequences

Exponential and Root Functions in Limits

Limit of (1 + 1/n)^n as n Approaches Infinity

Watch next