Sequences which converge to zero

This graph leads us to an important idea about sequences. We then say that zero is the limit or sometimes the limiting value of the sequence and write,. This notation should look familiar to you. It is the same notation we used when we talked about the limit of a function. Using the ideas that we developed for limits of functions we can write down the following working definition for limits of sequences. The working definitions of the various sequence limits are nice in that they help us to visualize what the limit actually is.

Just like with limits of functions however, there is also a precise definition for each of these limits. Now that we have the definitions of the limit of sequences out of the way we have a bit of terminology that we need to look at. So just how do we find the limits of sequences? Most limits of most sequences can be found using one of the following theorems.

This theorem is basically telling us that we take the limits of sequences much like we take the limit of functions. 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.

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. These properties can be proved using Theorem 1 above and the function limit properties we saw in Calculus I or we can prove them directly using the precise definition of a limit using nearly identical proofs of the function limit properties.

Next, just as we had a Squeeze Theorem for function limits we also have one for sequences and it is pretty much identical to the function limit version. This will be especially true for sequences that alternate in signs. The following theorem will help with some of these sequences. In this case however the terms just alternate between 1 and -1 and so the limit does not exist.

If we do that the sequence becomes,. In this case all we need to do is recall the method that was developed in Calculus I to deal with the limits of rational functions. We will need to be careful with this one. We will also need to be careful with this sequence. Also, we want to be very careful to not rely too much on intuition with these problems.

We will need to use Theorem 2 on this problem. Therefore, since the limit of the sequence terms with absolute value bars on them goes to zero we know by Theorem 2 that,. So, by Theorem 3 this sequence diverges. We now need to give a warning about misusing Theorem 2. Theorem 2 only works if the limit is zero.

If the limit of the absolute value of the sequence terms is not zero then the theorem will not hold. 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.

A sequence would be just e. Here you are dealing with a series. Don't confuse it with a sequence. Add a comment. Active Oldest Votes. Deepak Deepak 25k 1 1 gold badge 22 22 silver badges 50 50 bronze badges. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name.

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