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Numerade Educator

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Problem 11 Easy Difficulty

Test the series for convergence or divergence.

$ \displaystyle \sum_{n = 1}^{\infty} \left( \frac {1}{n^3} + \frac {1}{3^n} \right) $

Answer

convergent.

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Video Transcript

for this problem. We just need to recall that if you do not get something that is an indeterminate form, when you split up to some, then you are allowed to split up the sun. That what we mean that this can just be written as this class. This sum. Okay, so since both of these sums are convergent were allowed to do that. So this is just a finite plus finite equals finite? Yeah. Or, in other words, convergent plus convergent equals convergent. So if and if these sums were not convergent if instead we had, like, one over in plus one here and then we were subtracting one over in then this would not be a trick that we were allowed to dio. All right, could you definitely don't want to end up with something that's an indeterminate form. So if you are adding up to things that are virginity, you know it's possible that two wrongs make a right in that sense. So if we had won over n plus one and then we were subtracting one over in, we wouldn't be allowed to use the same type of trick, but with convergent, if it's just convergent plus convergent Then we can split it up like that and we get convergent Plus convergence is again convergent So we're convergence here.