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Numerade Educator



Problem 38 Medium Difficulty

Test the series for convergence or divergence.

$ \displaystyle \sum_{n = 1}^{\infty} (\sqrt[n]{2} - 1) $


The series is Divergent


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Video Transcript

for this problem, we're going to recall the fact that you, the exes one plus X plus x squared over two factorial plus X cubed over three factorial plus dot, dot dot and in the swearing of four is two. He is a number that's less than four. So the square root of E should be a number that's less than two. Therefore, this sum should be something that's bigger than the end through of the square root of E. In other words, the end through tiv e to the one half. No, another way of writing This is E to the one over two n. And now we can use this expansion up here on this, either one over to N to get any calls. One to infinity, one plus one over to end. Plus, I'm just going to put dot, dot, dot For these other terms, we started this minus one. So this one and this minus one will cancel these dot, dot, dot terms. The ones that we see here, they're going to be something that's positive. So whatever they are, we know that we're going to get more when we had them on. So this left hand side should be bigger than some from in equals, one to infinity of just one over to N. And this is just the harmonic. Siri's divided by two. So infinity, divided by two, is still infinity, so we will still get divergent.