# Sequences and convergence

Each of the following are equivalent ways of denoting a sequence.

### Sequence convergence test

Which would make us ask a very natural question-- what happens to a sub n as n approaches infinity? It's a function where we're limited right over here to positive integers as our domain. If we multiply it times negative 1 to the n, then this one would be negative and this would be positive. So we're going to get really, really, really, really small numbers. So just how do we find the limits of sequences? This right over here is negative 1. But this is still just a limit as something approaches infinity. In this case however the terms just alternate between 1 and -1 and so the limit does not exist. So we could write this as equaling negative 1 to the n plus 1 power over n. It is the same notation we used when we talked about the limit of a function. But let's say this is 1, 2, 3, 4, 5, and I could keep going on and on and on. And I'm going to graph y is equal to a sub n. However, it does tell us what each term should be. Gauss in his etude of hypergeometric series for the first time rigorously investigated under which conditions a series converged to a limit.

We want the first term to be positive. In this case however the terms just alternate between 1 and -1 and so the limit does not exist. And this is, let's say this right over here is positive 1. So, now that we know that taking the limit of a sequence is nearly identical to taking the limit of a function we also know that all the properties from the limits of functions will also hold.

If the sequence converges determine its limit. If the starting point is not important or is implied in some way by the problem it is often not written down as we did in the third notation.

### Convergent series

So, be careful using this Theorem 2. Each of the following are equivalent ways of denoting a sequence. This is an easy mistake to make when you first start dealing with this kind of thing. And I haven't rigorously defined it yet, but you can conceptualize what's going on here. So when n is equal to 1, a sub n is equal to 1. The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes. Well, let's think about if we can define a sub n explicitly.

The following theorem will help with some of these sequences. We start with a positive, then a negative, positive, negative.

## Convergent sequence example

Sequences of this kind are sometimes called alternating sequences. But this is still just a limit as something approaches infinity. As long as it-- if this was not equal to some number, then we would say that a sub n diverges. But if this is true-- so let me write this down. And so asking what the limit of a sub n as n approaches infinity is equivalent to asking what is the limit of negative 1 to the n plus 1 power over n as n approaches infinity is going to be equal to? A couple of notes are now in order about these notations. Now, I have not proved this to you yet. The last part of the previous example is a good example of this and in fact this warning is the whole reason that part is there.

Or another way of saying that is, what is the limit-- let me do this in a new color-- of a sub n as n approaches infinity? We will more often just treat the limit as if it were a limit of a function and take the limit as we always did back in Calculus I when we were taking the limits of functions.

However, it does tell us what each term should be.

So we see here that when n is equal to 1, a sub n is equal to 1.

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