Determine whether the sequence is convergent or divergent. If it is convergent, find the limit.

step 1 Hi guys, this is Kevin with Numerad. So let's consider a sequence that is denoted by 5 over 3 to

Determine whether the sequence is convergent or divergent....

Key Concepts

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index becomes very large. In practical terms, if the difference between the sequence terms and the limit becomes arbitrarily small for sufficiently large indices, the sequence converges to that limit.

Exponential Decay

Exponential decay refers to a process where a quantity decreases at a rate proportional to its value. In sequences, when a constant numerator is divided by an exponentially growing denominator, the fraction tends toward zero because the denominator increases much faster than the numerator.

Sequence Convergence

A sequence is said to be convergent if its terms approach a specific finite value as the index goes to infinity. In the context of analysis, checking convergence involves proving that the sequence gets arbitrarily close to a single number, known as the limit, as the index increases indefinitely.

Transcript

00:01 Hi guys, this is kevin with numerad.

00:03 So let's consider a sequence that is denoted by 5 over 3 to the end.

00:10 Now a sequence like this, we might be curious as to whether or not this converges.

00:13 So a nice way to begin to build our intuition is to maybe consider a table.

00:17 So let's begin with a table.

00:19 We'll do some n values, and then we'll call this a sub n.

00:23 A sub n is a common notation.

00:24 So we'll look at some a sub n values.

00:26 So let's look at a value of 1.

00:28 That's going to be 5 over 3.

00:32 We might then want to consider, since we are going to be looking at what happens as n goes to infinity.

00:37 And we can write that down here as a little bit of a side script.

00:39 So n going to infinity is the thing that we're curious about.

00:44 Let's not choose n equals two because that's not really capturing the idea that we're growing towards infinity.

00:49 Perhaps we grow it a little faster.

00:50 We say n value of 10.

00:52 Well, that's going to be 5 over 3 to the 10th power.

00:56 Now, 3 to the 10th power is quite a large number.

00:58 I don't know what it is up at the top of my head without pulling out my calculator.

01:01 But i can see that it's going to be significantly smaller than the value preceding it, because five divided by three is going to be much bigger than five divided by three to the ten, which is a much larger divisor.

01:14 So i've got smaller.

01:15 I'm kind of decreasing as i go across here...

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