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List the first five terms of the sequence.$ a_n = \frac {(-1)^{n-1}}{5^n} $

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$\left\{\frac{1}{5},-\frac{1}{25}, \frac{1}{125},-\frac{1}{625}, \frac{1}{3125}, \dots\right\}$

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.

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.

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List the first five terms …

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.List the first five terms…

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Write out the first five t…

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Write the first five terms…

first term. We're we're goingto have minus one to the one minus one. So minus one to the zero. Divided by five to the one. So anything that zero powers just one and five of the one is just five. So first term, it's one fifth. Next term we're looking at where plugging in n equals two. Now we're just gonna have minus one to the one divided by five. Squared A minus. I won over twenty five. Okay. And our terms should be alternating sign since we have minus one of the N minus one here. So this was positive. This was negative. Next term is going to be positive again. It'LL be minus one to the three minus one. So minus one squared, it will give us a positive number. Since his one over one hundred twenty five and drain equals four, you have minus one to the power of three. Which is the same thing is just minus one and we're dividing by five to four. So you can put that in a calculator in equals five. We should get a positive number since in equals. Forgive us a negative number, and this will be five to the five happening here. Okay, that's the first term. That's the second term. Third term for third, fifth time

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