In Exercises 37–44, find the sum of each infinite geometric serie...

Question

In Exercises 37–44, find the sum of each infinite geometric series.

3 + 3/4 + 3/4^2 + 3/4^3 + ...

Accepted Answer

everyone in this problem we are given an infinite series and were asked to evaluate the sum. Okay, in the series were given five plus 5/2 plus 5/2 squared plus 5/2 cubed plus dot dot dot. Okay now we look at this we can see that this is in the format of a geometric sequence or series. So let's recall the general format. We have some some From I equal zero to infinity of a. Times are to the exponent I Okay. It's gonna be that initial value in our is going to be our ratio between consecutive terms. Okay so that's our common ratio are Okay so let's start by trying to find our okay so let's find so again our is the common ratio between two consecutive terms. So let's take the first two terms. Okay the ratio between the second term And the first term. 5/2 divided by five. It's going to be 1/2. We can do the same between the next two terms. 5/2 to the exponent two divided by 5/2. This is going to give us one half again and you can do it again with the 3rd and 2nd term and you'll see that you get the same thing. Okay so we can go ahead and write are some as a sum from I equals zero to infinity. A Is the initial value. So that first value is five. Okay and then our our common ratio that we found one half to the exponent I Okay and you can check this. Okay when we have I is zero we get one half to the zeros. Once we get five so we get that first term. Then when I goes to one we have five times one half to the exponent one. That's five halves. Hey one eye is 251 half to the exponent to which is 5/ squared and so on and so forth. Okay so that does match what we're given. Okay And so are a value is five and our our value is one half. Now let's recall that the sum of an infinite series in this format is given by a over one -R. Okay This is going to be equal to five divided by 1 -1. This is equal to five divided by a half which is going to equal 10. Okay and so the some of this infinite series that we've been given is going to be 10. We have answer be that's it for this one. I hope this video helped see you in the next video

In Exercises 37–44, find the sum of each infinite geometric series. 3 + 3/4 + 3/4^2 + 3/4^3 + ...

Key Concepts

Infinite Geometric Series

Sum of an Infinite Geometric Series

Convergence of Series

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