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JH
Numerade Educator

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Problem 10 Easy Difficulty

Use the Ratio Test to determine whether the series is convergent or divergent.

$ \displaystyle \sum_{n = 0}^{\infty} \frac {( - 3)^n}{(2n + 1)!} $

Answer

The series is absolutely convergent.

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AC

Alexander C.

May 13, 2021

how the fuck did you get +3???

Video Transcript

let's use the ratio test to see whether or not the Siri's converges. So let's to note this by an now. The ratio test requires that we look at the limit and goes to infinity a flu value a n plus one over an. Now let's go ahead in the numerator and plus one. All right, here will have to in plus three than in the denominator. We just have a n no. All right, so here, let's go in and simplify this. We could cancel and of these threes here these negative threes on one left over. It's absolute value so that'LL just become three and then this will come up to the numerator and then we still have this in the denominator. Select wants the next patient because I'm running out of room. What? This was our limit. Now we can go ahead and rewrite this in the denominator weaken right, this is too, and plus one factorial and then two in plus two, two and plus three. And then we could cancel off the two and plus one pictorial. And as we take that limit, we have three up top and the denominator goes to infinity. So this is equal to zero because this is three over. Infinity equals zero. The women is general. This is less than one. So we conclude the original series on page one. Good. And write that out. Factorial. This one converges by the ratio test, and that's our final answer.