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For each geometric series determine and r. If the series converges, find its sum. If the series diverges , explain why:(4 pts)(4 pts) 5# (T)"pts) 81 _ 27+9- 3+...

Question

For each geometric series determine and r. If the series converges, find its sum. If the series diverges , explain why:(4 pts)(4 pts) 5# (T)"pts) 81 _ 27+9- 3+1-3 5 +.=

For each geometric series determine and r. If the series converges, find its sum. If the series diverges , explain why: (4 pts) (4 pts) 5# (T)" pts) 81 _ 27+9- 3+1-3 5 +.=



Answers

Use the formula for the sum of a geometric series to find the sum or state that the series diverges.
$$
5-\frac{5}{4}+\frac{5}{4^{2}}-\frac{5}{4^{3}}+\cdots
$$

This coverage is our diverges. So let's go ahead and simplify what we have here. That's four to the negative are Yeah. Plants and they are over for today. Are can intent sympathize too five or four. They are power here. It's taking that from our equals one to infinity here. Okay. So that we can see this inner term here, absolute value thereof, Single to five or 4. So that is definitely greater than one. Anytime this happens there is greater than or equal to one. You know that this series that purchase. So that's her answer there.

In this question will be calling about the geometric experience which, under form submission in the constant eight times are about an from zero to infinity and this is equal to the first term. The venue by one minus are even only if the absolutely are must be strictly smaller than one. It will be divergent even only if the absolute are must be quite equal to one in this question were given for about three over 5003 breast for about 4/5 pound four. That's for about five over 5005. And so, um, take the ladies when we can understand into the threes on the farm for over five hour and and start from three up to infinity So clearly we can identify. This one will be the geomagnetic Siri's with the absolutely the r equals, you know, for over five smaller than one. Therefore by formula, we get equal with you using this formula, we see the first time here echoes uniform 3/5 3 and then minus one minus. There are he will be for over five. Then we get Nico to the for About three will be a co 64 over 125 over this one will be five minutes for Could be won over five. So we get the answer it we go to the 60. 64 off. Nah. Uh, 25. And that's gonna be the answer.

Okay. This question gives you a series 5 -5 Fords plus 5/4 squared minus 5/4 cubed. And if I take the ratio of each term to the term before it, I can tell that I'm multiplying by negative 1/4 every time. So the common ratio yeah equals negative 1/4. The absolute value of that common ratio Is less than one. So this infinite geometric series will converge converges to a 1/1 minus r. A one is the first term. Five Divided by 1 -3 common ratio, negative 1/4. His five divided by 5/4 which is four. The sum to infinity is positive for. And that's it for this problem. This problem comes up, comes out kind of quick. If you're not sure about the common ratio, remember that I can always do the second term divided by the first and then the third term divided by the second and so on. Oops hold on, I still get negative 1/4. And that needs to be a pattern that you can tell continues. Or in general I could take a sub N plus one, divided by a sub N. Mhm. If that number is a constant for the problem, then that's your common ratio. As long as the absolute value of your common ratio is less than one. Then I can use this formula, but only if your ratio is less than one. Be careful about that.

Okay for this question, I'm given 4/3 over 5/3 plus four. Four thirds. I'm sorry for four to the third over five to the third. For the 4th over 5 to the 4th. Four to the fifth over five to the fifth dot dot dot. This is my first term, my second term in my third term, my 10th term looks like 4 to the N -2, Divided by 5 to the in minus two. Or I could say it's 4/5 To the N -2. So I'm looking at the sum of 4/5 To the N -2. I'm sorry, in plus two. In plus two when in equals three, The exponent is 2-plus 3, which is five and so on. So I need to first decide if this is a geometric series and it is a geometric series. If I'm multiplying by the same thing. Every time any time I have a number raised to the end, a constant number Raised to the 10th power, then that constant number inside is going to be your common ratio. I can also find it by finding a three over a two, a two over a one and making sure they always equal the same thing. When I divide I get 4/5 each time. So the sum Of an infinite geometric is a 1/1 -2. If the absolute value of our is less than one, So are is 4/5 So 4/5 is less than one. So I take the first term for cubed over five cubed, Divided by 1 -3 Common ratio. So that is four cubed over five cubed Times the reciprocal of the bottom, which is 1/5. That's 5/1. So I get four cubed divided by five squared. That is my final. Some thanks for watching.


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