Problem 13 Easy Difficulty

Determine whether the series converges or diverges.
$ \displaystyle \sum_{n= 1}^{\infty} \frac {1 + \cos n}{e^n} $

Answer

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Video Transcript

Let's first note that the terms that we're dealing with their positive. So first, let's look at the one plus co signed in. We know that CO sign of end is less than or equal to one favor than or equal to negative one. So if we add one so all signs of this inequality here we have zero less than or equal to one plus coz I less than or equal to two. So that shows that our numerator is positive and we know he's always positive. The reason I'm checking this is because if you like to use the comparison test, you have to make sure that your Siri's has on ly positive terms. And that's what we have here. A n bigger than zero or equal to That's fine, just no negatives. So now let's go ahead and use comparison his test here. So I know one plus co sign in is less than or equal to two. So this tells me that our Siri's is less than or equal to to overeat of the end. All I'm doing here is just using this inequality that one plus coastline is less than or equal to two and then we can rewrite this. Pull out the two and then we could write. This is one over e to the end. This is a geometric Siri's. We see that our equals one over e rough estimates of this would just be a third. But all that matters is that it's less than one an absolute value, one over three, more or less, and that's less than one. So any time it's geometric series. Satisfied this? We know that it converges. Therefore, since we have a Siri's with positive terms and it's founded above by a convergence here ese by the comparison test our series, which is one plus co sign and over eat of the end. Also convergence okay, and that's your final answer.