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# Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.$\displaystyle \sum_{n = 1}^{\infty} 12(0.73)^{n-1}$

## $$\frac{400}{9}$$

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##### Catherine R.

Missouri State University

##### Samuel H.

University of Nottingham

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let's determine whether this geometric series is conversion or divergent, and if it's conversion, will go and find the summer as well. So recall geometric Siri's of this form, then hear this converges. If the absolute Value bar is strictly less than one sonar problem, we see our his point seven three, and the absolute value of that is just point seven three. That's less than one. So this converges that answers the first question. So come up here we'LL see. It's convergence, not divergent. And now, for convergent geometric series, we have a formula for the sun and equals one twelve, and we have our our point seven three to the end minus one. So, of course, here, when I wrote the Geometric series, it doesn't matter whether you have n here and minus one. It's still geometric. So in this case, the formula is we always take the first term of the entire series that will correspond to plugging in the starting number down here, the first number and equals one into an over here, and I don't give you the first time and then one minus R. So if we go ahead and plug in and equals one. We have twelve point seven three to the one minus one. That's point seven three to the zero. Power is just one, so we could ignore that term and then one minus point seven three. So I'm getting twelve over point two seven. This can be simplified, too. See twelve over twenty seven over one hundred. And that simplifies too. Four hundred over nine. And that's our final answer. This's the sum of the geometric Siri's.

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