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Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)$ a_n = \frac { 1 \cdot 3 \cdot 5 \cdot \cdot \cdot \cdot \cdot (2n - 1)}{n!} $

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divergent

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

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Lectures

01:59

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

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In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

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Use a graph of the sequenc…

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Let's use a graph of the sequence to decide whether converges or die rouges. So here, let's use this formula to find some of the ends. So here, for example, when we plug in and equals one, we just multiply up, too. Just one itself, divided by one factorial equals one. Now a two that's one times two times two minus one over to factorial. So three halfs and then a three similarly fifteen over six. So it's a little bigger than two a. Four. This is one oh, five over twenty four. Now that's a little bigger than four. So not only assisting increasing, but it seems the rate at which it's increasing is increasing, and one leads to another term here, a five that's nine forty five over one hundred twenty. A rough estimate of this more or less his nine Less than that, maybe eight would be closer. So definitely much bigger than before. So a little bigger than here around, eh? So we could see that this thing is just increasing, and this is just diversion is growing too fast. So from the graph we decide that it's diversion, and because of that, there's no more works show. So you want a little more insight here. You could not only is a diversion, you could even say the limit of Anna's infinity, and that's our final answer.

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