When is a sequence said to be bounded

In this case, we write. To be more precise, we now present the more formal definition of limit for a sequence and show these ideas graphically in Figure.

If a sequence does not converge, it is a divergent sequence, and we say the limit does not exist. As defined above, if a sequence does not converge, it is said to be a divergent sequence. However, different sequences can diverge in different ways. The sequence is, in fact, divergent.

Writing that the limit is infinity is intended only to provide more information about why the sequence is divergent. A sequence can also diverge to negative infinity. Because a sequence is a function whose domain is the set of positive integers, we can use properties of limits of functions to determine whether a sequence converges. We now consider slightly more complicated sequences. The algebraic limit laws allow us to evaluate limits for many sequences.

In the next example, we make use of this fact along with the limit laws to evaluate limits for other sequences. For each of the following sequences, determine whether or not the sequence converges. If it converges, find its limit. We conclude that. This idea applies to sequences as well. This property often enables us to find limits for complicated sequences.

From Example a. Continuous Functions Defined on Convergent Sequences. Another theorem involving limits of sequences is an extension of the Squeeze Theorem for limits discussed in Introduction to Limits. Use the Squeeze Theorem to find the limit of each of the following sequences.

Using the idea from Exampleb. Here is a summary of the properties for geometric sequences. Home Calculus Sequence and Series. Still Confused? Nope, got it. Play next lesson. Try reviewing these fundamentals first Introduction to sequences.

That's the last lesson Go to next topic. Still don't get it? Review these basic concepts… Introduction to sequences Nope, I got it. Play next lesson or Practice this topic.

Play next lesson Practice this topic. Start now and get better math marks! Intro Lesson: a. Intro Lesson: b. Lesson: 1a. Lesson: 1b. Fold Unfold. Bounded Sequences of Real Numbers. Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.

Click here to edit contents of this page. Click here to toggle editing of individual sections of the page if possible. In the previous section we introduced the concept of a sequence and talked about limits of sequences and the idea of convergence and divergence for a sequence.

In this section we want to take a quick look at some ideas involving sequences. In other words, a sequence that increases for three terms and then decreases for the rest of the terms is NOT a decreasing sequence!

Also note that a monotonic sequence must always increase or it must always decrease. In other words, there are an infinite number of lower bounds for a sequence that is bounded below, some will be better than others. Also, since the sequence terms will be either zero or negative this sequence is bounded above. Therefore, while the sequence is bounded above it is not bounded.

The sequence terms in this sequence alternate between 1 and -1 and so the sequence is neither an increasing sequence or a decreasing sequence. Since the sequence is neither an increasing nor decreasing sequence it is not a monotonic sequence.

The terms in this sequence are all positive and so it is bounded below by zero. Therefore, this sequence is bounded. The sequence is therefore bounded below by zero. Likewise, each sequence term is the quotient of a number divided by a larger number and so is guaranteed to be less than one.