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JH
Numerade Educator

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Problem 22 Easy Difficulty

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {5}{\pi^n} $

Answer

The geometric series converges and the sum is $\frac{5}{\pi-1}$

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EG

Ethan G.

November 16, 2020

what test or method were you using?

Video Transcript

interment weather this geometric Siri's converges. And if it does converge, let's go ahead and find the song. So this Siri's We could go ahead and rewrite this to see if it's geometric. Well not. Not only we know it's your metric they told us that, but if we go ahead and rewrite this as one or a pie to the O. R. Teo, excuse me to the end. We now have it in the form a r n. And so here we can use the formula. So the formula is you take the first term in the sun and then you divide it by one minus are the common ratio. Hear our case, our equals one over pi. Then I think about this formula here is it doesn't matter what you're starting value for end is and it doesn't matter what the exponents is, for example, could be and plus one and minus three. This formula will always work for geometric assuming, of course, we only use this when it converges. And that's when absolute value are is less than one equivalently negative, one less than are less than one in our problem. This value satisfies that it's positive so it's bigger than negative one, but it's more or less one over three, so that's definitely less than one. So this will converge. And then if we use the formula while the first term here the first term, when is when you plug in and equals one? See you get five over pi and then we have one minus R so one minus one over pi And if we like, we could just go ahead and simple by this five over pi up top and then on the bottom, pine minus one over pi and we end up after cancelling those pies. We have five time itis, one on the bottom and that's your final answer.