Express each sum using summation notation. (2)/(3)-(4)/(9)+(8)/(27)-⋯+(-1)^12((2)/(3))^11

Name: Problem 64, Chapter 11: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2020-06-21T20:48:31-08:00
Duration: 2 min 33 s
Channel: Julie Silva
Description: Express each sum using summation notation. (2)/(3)-(4)/(9)+(8)/(27)-⋯+(-1)^12((2)/(3))^11

step 1 We're being asked to write the given sum using summation notation. So the first thing we're goin

Express each sum using summation notation....

Key Concepts

Summation Notation

Summation notation is a concise mathematical way to represent the sum of a sequence of terms using the Greek letter sigma (?). It involves specifying an index variable, its range, and a general term. This notation is particularly useful for capturing series with many or infinitely many terms in a compact form.

Geometric Series

A geometric series is a series where each term is obtained by multiplying the previous term by a constant factor known as the common ratio. This property allows the entire series to be expressed in terms of the initial term and the common ratio raised to increasing powers, which simplifies the analysis and computation of the sum.

Alternating Series

An alternating series is a series in which the signs of the terms alternate between positive and negative. This alternation is often represented by including a factor such as (-1)^n or (-1)^(n+1) in the general term. The alternating structure can have significant implications for the convergence properties of the series.

Transcript

00:01 We're being asked to write the given sum using summation notation.

00:04 So the first thing we're going to do is find the formula that will help us find any term in this given sequence.

00:09 And to do this, we're going to use our very last term.

00:13 So it looks like we have our pattern negative 1 to sum power, which is going to account for the audit every other term being positive or negative.

00:20 And then, as you notice, we have two thirds getting raised to sum power, which if you notice the pattern for your terms, it is two thirds raised to some power.

00:28 So the key thing here is we're going to have to write our.

00:30 Formula, but our exponents have to be in terms of k, depending on what term in the sequence it is.

00:37 Well, because two -thirds is our first term, i'm going to want two -thirds in this first term being raised to the k power.

00:43 So i'm going to have two -thirds raised to the k power.

00:48 So now, how are we going to express our first one? well, negative one raised to the, well, if 11 is our k, we're adding one to get from 11 to 12.

00:57 So this would be to the k -plus -1 power.

01:00 So now we've written the formula to find any term in the sequence.

01:04 So now we have to figure out our values for k.

01:06 Well, what value for k would make our first term two -thirds? and that would actually be when k is equal to 1.

01:13 So this is going to start at k equals 1...

Please give Ace some feedback