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JH
Numerade Educator

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Problem 60 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} (-4)^n (x - 5)^n $

Answer

Interval of convergence is $x \in\left(\frac{19}{4}, \frac{21}{4}\right)$
The sum of the series is $\frac{1}{4 x-19}$

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Video Transcript

Let's go in and find the X values for which this Siri's converges that'LL be our first step. And then for those values of X that we get in part one. Well, go ahead and also find the sum of the Siri's so that'LL be our second step. So for the first step, the first thing I would do here is to rewrite this, since it's actually and geometric Siri's but kind of in disguise the way that it's written. So rewrite this by pulling out the N and the reason I could do that. I'm just using this fact here that you've seen before. You're laws of exponents. You could always do this. So I'm just using this fact with a equals minus four. B equals X minus five. So I see that this is geometric and I see that my our value, the thing that's being raised, the end power is negative four times X minus five. And then I know that a geometric series will converge, huh? If the absolute value bar is less than one, so in our case, we have absolute value, are so it's negative for parentheses, X minus five, and this all inside the absolute value. And so that's not a wonder that's absolute value of make out a little larger now. I could simplify this, said it less than one and then software X. So we have Let's divide by four and then using the definition of these absolute values here we have. And then at five all sides here five. Which you could raise twenty over four so have nineteen over for less than X, less than twenty one over four. So that's the answer for Part one, The's Air the Axis, for which the Siri's convergence now for Part two will assume that X is in this interval here. So that a convergence and now we'LL find the sum. So the sum for geometric series is always the first term of the Siri's. What you get by plugging in the starting number here and equal zero for N over one minus R. Now, in our problem, if we plug in and equal zero, we get zero exponents was So we just get a one in the numerator and then in the denominator we have one minus are and we know that our equals negative for Times X minus five. So watch out for that double negative there. So now and the denominator, we have one. And then what are we left over with? So how is one going to combine? So we have a plus here That's a plus for but then were multiplying. It's in negative five, so let's negative twenty. But we have a plus one here, so that's negative. Nineteen. But we also have positive four times X, so there's a for X there as well. So assuming that X is between nineteen over four and twenty one over, for the sum of the Siri's that was given to us is equal to one over for X minus nineteen, and that's your final answer.