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JH
Numerade Educator

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Problem 61 Medium Difficulty

Find the values of $ x $ for which the series converges. Find the sum of the series for those values of $ x. $
$ \displaystyle \sum_{n = 0}^{\infty} \frac {2^n}{x^n} $

Answer

$$\frac{x}{x-2}$$

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Video Transcript

Let's find the X values for which this Siri's that's given converges. So that would be the first part of the question here. And then the second task is for those values of X. In Part one, let's go and find the sun. So for part one, let me rewrite this, since it's actually a geometric series. But it might not be obvious by the way, that it's running right now, so I pull out the end outside of the parentheses and the reason we're allowed to do that. That's just the law of exponents here. That's just eight of the n over B to the end. So I'm just using this fact here with two equals A and X equals B. So I see that this is geometric. I see that my our value is two over X, and I know that a geometric Siri's will converge on ly if the affluent value our is strictly less than one. That's the requirement for convergence for geometric series. So in our case, we have absolute value. Are a flute value to over X? Let's always simplify that before setting it less than one. So all right, this is too over absolute value X, since two is always positive, less than one and then go ahead and solve this inequality. So let's multiply both sides by absolute value X, and here we can use the definition of the absolute value. Tto find an inequality for X or possibly more than one inequality. So in this case, we have the access to be bigger than two or eggs has to be less the negatives who so depending on how you want to write your answer for part one here, you could also write. This is negative. Infinity to negative, too. Oh, and the union for the or and then to to infinity. So either of these for number one is acceptable. And then for Part two, Let's go out and save some room. Here, make some room on the side for part two, assuming that X is in this interval here, so X is bigger than to, or less than negative, too. Now, let's go ahead and find the sum. So since this is geometric, the formula for geometric this sum is equal to it's always the first term. So this thing right here is our some first term over one minus our common ratio. The first term year happens when you plug in and equal zero. Or you could even do it over here. If you plug in an equal zero, you get to over X to the zero power, which is one. And then we have one minus R, which is two over X, and then you could simplify this to be X over X minus two. So that will be the sum of the Siri's, and that's our final answer.