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JH
Numerade Educator

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Problem 34 Easy Difficulty

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {2^n + 4^n}{e^n} $

Answer

Diverges
Hint: This is sum of two geometric series

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Video Transcript

let's determine whether or not the Siri's conversions. So here, If we notice that four over and over either then you can rewrite this. And if we take the limit, this will go too Infinity as n goes to infinity since four over e is bigger than one room is about two point seven. So if you take a number larger than one and keep multiplying it by itself and you go to the limit, this will go to infinity So we could see that because of this Since this is bigger than or equal to for the end over either then and we just showed that this term goes to infinity. Since this is larger, this must also go to infinity in the limit. So we have lim and goes to infinity to end put Teo and plus four the end over either of them. And this will equal infinity and we use the test for diversions. So I hear you're given a Siri's and your terms or an so if the limit of am does not exist or if the limit does exist. But it's not equal to zero, which is our case Here we have infinity This is not equal to zero, then the Siri's will diverge. So our case, we took the limit. We're using the test for diversions. We took the limit of our way in here. Delimit exist, but it's not equal to zero. Therefore, by this test for diversions, our Siri's will diverge. So I should put diverges, and that's a final answer.