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JH
Numerade Educator

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Problem 62 Hard 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} \frac {\sin^n x}{3^n} $

Answer

Interval of convergence is $(-\infty, \infty)$
The sum of the series is $\frac{3}{3-\sin x}$

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Video Transcript

Let's go ahead and find the values of X, for which the Siri's converges That's one park. And then for those values of X for which it converges in part one will go out and actually find the summers. Well, that'LL be part two for per one. This is actually a geometric Siri's here, and it's a little easier to see that if you just first write the fraction and then write the exponents outside. Oops, that's not a three out there. That should be an end. And the reason you're allowed to do this is just coming from your laws of exponents a n over being. So here I am just using a equal sign and then be equals three. So if I write it this way, it's easy received at its geometric and that are equals sine X over three. And I know that a geometric series will converge if the absolute value are is less than one. So in this case we need we have absolute value, are which is the absolute value of sign over three. Let's simplify that before we said it less than one. Now let's set this less than one and try to solve this for X. So let's multiply that three to the other side. And this inequality here is always true because we know that always true since we know that sign is always between negative one and one. So if you're always between negative one and one than an absolute value, you're less than or equal to one. And therefore you have to be less than three. So this inequality is true. Oh, for all ex. So here, if we want, we can say negative infinity less than X less than infinity. That's the answer to the first part of the question. All real numbers, all real numbers. That's another way to say the same thing here. That's the interval notation. The inequality notation Negative infinity less than X minus infinity. That's the same thing is saying just all real numbers now for part two. We actually want to go out and find this sum for any value of X that you plug in. We want to find the corresponding some. So we're evaluating this after you pick X so too. Since this is geometric, I know that the sun will always equal the first term and then I divide by one minus are the common ratio. So in this problem, the first term when you plug in and equal zero here you have signed over three. But to the zero power that's just going to give you a one back. And then you have one minus R, which is signed over three. And this thing over here, if you want, you could go ahead and rewrite this as three over three minus sign X. So the way to do this is just tow, get a common denominator down here and then just simplify things. In either case, the Siri's converges for all roll numbers X and for any role, our next that you use this down here in the bottom left is the value of the sum, and that's your final answer.