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JH
Numerade Educator

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Problem 40 Easy Difficulty

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \left( \frac {3}{5^n} + \frac {2}{n} \right) $

Answer

Divergent

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Video Transcript

Let's determine whether this Siri's is conversion of average and if it convergence will go ahead and find the summer as well. So before we look at this some here, let's just look at it. There's some over here while this If this did converge, we would be able to write two times the summation. However, this new series here that I just wrote, This is what your bull calls the harmonic series. So see example, nine in the textbook. And this one is diversion. Sophie multiplied by two. It's still diversion. So this one will be right here will be that version as well. Now we have that this series here is that version. What does that tell us about the original one? Well, if this one did converge, we would be able to write it as and equals one infinity three over five in. Take a step back there plus two and then one over end. However sent due to the second term here, we just showed that this was diversion. There's some over here. On the other hand, this one actually is conversion. It's geometric. And here you see, you can write three over five to the end is three times one over five to the end. So our equals one over five, and any time you add a conversion plus a divergent series, while the result will be a diversion, Syrians therefore our syriza's divergent and as a result, we do not have to find a son because it doesn't exist, and that's your final answer.