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JH
Numerade Educator

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Problem 33 Easy Difficulty

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{4 + e^{-n}} $

Answer

$$\frac{1}{4} \neq 0$$

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Video Transcript

let's determine whether or not this series converges. So conversion or diversion, and if it ends up being conversion, we will have some more work to do. We'll go ahead and find that some as well. So here, this doesn't look like geometric. So maybe the tests that I would use first here is what's called the test for divergence. Yeah, so this requires that we look at the limit as n goes to infinity of 1/4 plus and let me go ahead and rewrite. This term is one over e all to the end. Okay, now let's go ahead and take that limit. Since one over e satisfies the following inequality, the limit as n goes to infinity of one over E to the end will equal zero. So as we take the limit, this term here goes to zero and we're just left over with 1/4 plus zero equals 1/4. However, this shows that the limit of the terms in the series is 1/4. So it exists, but it's not equal to zero, and this is where we can use the test for the divergence. So since the limit as n goes to infinity of 1/4 plus, even the minus end exist but is not equal to zero. The series diverges by the test for divergence, and that's our final answer. Yeah, mhm.