Problem 15 Medium Difficulty

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {\sqrt n + 4}{n^2} $

Answer

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Video Transcript

to determine the convergence of this Siri's I could be helpful to split it up. So we have the sun for many girls. Want to infinity? Ah, in the squared event plus four over in scored. I'm going to split that up like so? Because if we split up the sun and show that each piece converges, then the overall sum itself will converge. So since we're summing sums, we can split these up. Well, you know, one son of just squared of n over and squared Nowthe squared of end over and squared is scenting his end of the one half over and squared, which is one over and two minds one half which is into the three halves. So we have won over and three halves plus the sun from an equals one to infinity for over and squared. And since this is a four over and squared, sorry could be equal to four times one over and squared. We can split that up as well. So we end up with the son from an equals one to infinity, one over n to the haves plus four times the sun and equals one insanity of when over and squared. And now bye Pee Siri's We have won over end to the P P greater than one. That means that the Siri's converges So the first son converges. The second son also converges Pipi, Siri's and the sum of two conversion. Siri's well itself be convergent, so our overall Siri's is convergent.