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Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.$ \displaystyle \sum_{n = 1}^{\infty} 12(0.73)^{n-1} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Missouri State University

Campbell University

University of Michigan - Ann Arbor

University of Nottingham

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.

02:28

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.

03:23

Determine whether the geom…

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Determine whether the seri…

let's determine whether this geometric series is conversion or divergent, and if it's conversion, will go and find the summer as well. So recall geometric Siri's of this form, then hear this converges. If the absolute Value bar is strictly less than one sonar problem, we see our his point seven three, and the absolute value of that is just point seven three. That's less than one. So this converges that answers the first question. So come up here we'LL see. It's convergence, not divergent. And now, for convergent geometric series, we have a formula for the sun and equals one twelve, and we have our our point seven three to the end minus one. So, of course, here, when I wrote the Geometric series, it doesn't matter whether you have n here and minus one. It's still geometric. So in this case, the formula is we always take the first term of the entire series that will correspond to plugging in the starting number down here, the first number and equals one into an over here, and I don't give you the first time and then one minus R. So if we go ahead and plug in and equals one. We have twelve point seven three to the one minus one. That's point seven three to the zero. Power is just one, so we could ignore that term and then one minus point seven three. So I'm getting twelve over point two seven. This can be simplified, too. See twelve over twenty seven over one hundred. And that simplifies too. Four hundred over nine. And that's our final answer. This's the sum of the geometric Siri's.

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