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Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {(-3)^{n -1}}{4^n} $

$$\frac{1}{7}$$

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Let's determine whether this geometric Siri's converges or diverges and if it converges, will go ahead and find the sun. So first, let's recall Geometric series is typically written in the following form and we know that this will converge on ly when I flew. Value are is less than one, otherwise it averages so otherwise. But I mean by that is the absolute value bar is bigger than or equal to one and in our case, in diverges So this is for geometric So first, let's go ahead and rewrite are some so that it looks in this form so we could say what our is. So what I'll do there is I'm just going to play around with the numerator I like a three two that I prefer three did and power are negative three then power So what I'LL do here is I'Ll just multiply divined by negative three That gives me negative three the end over negative three So I have negative three to the end over negative three and the numerator and I'm still dividing by four to the end and then here because these air both to the end power let's go ahead and pull out the fraction. So that would be negative. Three over four to the end and then were multiplying this by negative one over three. So we can see that are are the thing that's being raised to the power equals negative three over four. It satisfies this which is less than one so and is conversion and purchase. Using this fact appear converges when the absolute value are is less than one three over for less than one so converges. And in this case, we also have a formula for the sun. So let's write that out and equals one to infinity. Negative three over four to the end. Negative one, sir. So the formula says you take the first term and then you divide by one minus R. So in our case, the first term is the one that you get by playing in the first value, and down here in our case happens to be one. So plug in one friend. So you get negative on the top. You get negative three to their one that'LL cancel with this minus three. So you just get one over four in the dim writer when you plug in a and equals one. And then you have one minus our value of our, which was negative, three over four. So this right here, we can go in and simplify that. Let's just go to the next page to write this. So we had one and they canceled that double minus. So that's one over four. And then we have seven over four. Selection's cancel most force to get our final answer of one over seven. So the Siri's conversions and won over seven is the value of the Siri's.