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JH
Numerade Educator

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Problem 57 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 = 1}^{\infty} (-5)^n x^n $

Answer

$$\frac{-5 x}{1+5 x}$$

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Video Transcript

Let's find the values of X for which this Siri's converges and for those values of X for which it does converge, we'LL actually find that some of the Siri's so there's two parts here. So for part one, let's first find out when this thing will converge. So here I can rewrite this sum. This is actually geometric series here because you, Khun, pull out the end. All amusing here is the fact that eh bien equals a nbn some going from this direction to that direction. So I pull out the exponents and then I see that this is geometric and I could see that they are is negative five x And I know that geometric will converge if the absolute value our sonar keys, the absolute value of negative five X We're just five Absolute value of X is less than one. So we need the condition. Absolute value are strictly less than one not equal to one. And this is precisely why I use this inequality at the very end to set it less than one. So song that Rex, by dividing both sides by five and if you want, you could go ahead and use the definition of the absolute value to rewrite. This is an apology. So these are the values of X for which this geometric series will converge. So that's the first part of the question. So let me try to save some space here. So that's part one now for part two, using for those values of X. So the ones that we got in part one of the ones in which it actually converges now we assume X is between negative one over five and one over five and now sensitive convergence. Let's find what the some of the Siri's is. So we know for geometric the sun well, equal. It's always the first term of the Siri's, so this formula works for GE Arliss of what? The starting point for enemies over one minus. R. So in this problem, the first term of the series comes from plugging in and equals one. So we have a negative five x to the one power up top over one minus R. But our was negative five x So we just have negative five X in the numerator and then one plus five x in the denominator. That's our final answer. So the Siri's conversions for these values of X over here and for any of those values of X, the sum of the entire Siri's is given by this expression over here in part two.