55–70. More sequences
Find the limit of the following se...

Question

55–70. More sequences

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

{nsin³(nπ / 2) / (n + 1)}"

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says find the limit or determine divergence of the sequence CN, which is equal to N, multiplied by sine, raised to the 5th power of the quantity of n pi divided by 2 in quantity, divided by the quantity of n + 1. So we're going to begin by looking at the sine term in our expression. So we were given sine to the 5th power of the quantity of N pi divided by 2 in quantity. Now this quantity actually cycles through the values of 010, and minus 1 in that order, since in here is an integer value. So when N is equal to 0, we get the sine to the 5th power of 0, which is going to be 0. When N is equal to 1, we get the sign to the 5th power of the quantity of pi divided by 2 in quantity, which is equal to 1. When N is equal to 2, we get the sign to the 5th power of pi, which is 0, and when N is equal to 3, we get the sign to the 5th power of the quantity of 3 pi divided by 2 and quantity, which is equal to -1. So this means if we look at our sequence, that CN is going to be equal to 0 for even values of N. And CN is going to be equal to the quantity of plus or minus 1 in quantity multiplied by N divided by the quantity of N plus 1 in quantity for odd values of N. Now, we're interested in the case as N approaches infinity. So if you look at the limit as N approaches infinity of N divided by the quantity of N + 1 in quantity, we see that this is going to be equal to 1. So that means for large values of N that C of N is going to be equal to -1 or it's gonna be equal to 1 for odd values of N. So, for large values of N, Basically, as in, Approaches infinity. CN is going to oscillate. Between three values, so CSpin oscillates. Between The values of -1, 0, and 1. And since our sequence here is oscillating between these three values, it never converges. So that means that this sequence actually diverges, which is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.

{nsin³(nπ / 2) / (n + 1)}"

Key Concepts

Limits of Sequences

Behavior of the Sine Function at Multiples of π/2

Sequence Divergence and Oscillation

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