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JH
Numerade Educator

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Problem 4 Easy Difficulty

Calculate the sum of the series $ \sum_{n = 1}^{\infty} a_n $ whose partial sums are given.
$ s_n = \frac {n^2 - 1}{4n^2 + 1} $

Answer

$\frac{1}{4}$

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Video Transcript

Let's calculate the sum of the Siri's here. So by definition, first lesson's rewrite this as a limit. Let's actually is a different letters since and is being used. Let's say our goes to infinity and then we have our son from an equals one of Tau are And then, by definition, I could go ahead and replace this with S R. This is the Arth partial some This is the notation at the author uses in the section for the partial sums as his talent. It stands for the partial Some If this sub script are is telling you how many terms of this a sequence that you want to add up And if you started a one and you always go up to the index here, so up they are. So this is just limit as our goes to infinity of S r. And then this is where we can use our formula for the esses. So here we're just using our instead of end again. The only reason I'm not using and here under this is because they decided to already use and over here, so replace as and are as our with R squared minus one over four R squared plus one. Now, you may recall many several ways to go around about this limit here, so you will want to be familiar with your limits. You could do open house rule here because the numerator and denominator we're both going to infinity. So if you'd like or this case, you could actually just go ahead and divide. It's happened bottom by the highest power of armed, the bottom civil divides happened. Bottom by R squared so top. That's what we give on the bottom. That's what we get. And as we take the limit, our goes to infinity. Those go to zero and we're left with one over four, and that's your final answer.