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Determine whether the series is convergent or divergent. If it is convergent, find its sum.$$ \displaystyle \sum_{n = 1}^{\infty} [(-0.2)^n + (0.06)^{n - 1}] $$

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The series converges to $\frac{7}{3}$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Campbell University

Harvey Mudd College

Idaho State University

Boston College

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.

01:50

Determine whether the seri…

02:46

01:14

Determine whether the geom…

03:17

01:16

01:55

01:30

03:49

02:31

06:21

let's determine whether this Siri's over here is conversion or diversion. If it commercials, then we'LL go and find the song. So before we look at the Siri's if you look inside the Siri's, you see a sum here. So let's just look at the two Siri's individually and see whether these converge the's air. Both geometric for this one. Over here, we see our is negative. Zero coin, too. Here, ours Cyril point zero six. And we know that a geometric series Comm urges. If that's Lou, value are is less than one. Since both of these are ours will satisfy that both Siri's will converge So by both. I'm referring to these two. And now we're just using a result that says that if these two theories individually converge, then you can go ahead and take the summation decimation outside and distributed to both terms. So here we have to separate geometric series. We know they both converge, really, because we looked at the our values over here. And, you know, the sum of a geometric series is one not one first term of the Siri's over one minus R. So in this problem, for the first time, if you plug in and equals one, that's your first term and then one minus R. And now for the second first term, plug in and equals one. You get this number here, This decimals with zero power, which is one and then one minus your are such is using this formula two times. Now, let's just go to the next page to simplify this last term down here so I can write. This is and then I could simplify this denominator. Six over five, then plus one hundred over ninety four. So here and this is after cancelling those fives there. And then we find that we have well, two, five, three over to a two. And this is our final answer. This is the sum of the conversion series.

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