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Numerade Educator

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Problem 74 Hard Difficulty

Express the limit as a definite integral.

$ \displaystyle \lim_{n \to \infty} \frac{1}{n} \sum^{n}_{i = 1} \frac{1}{1 + (i/n)^2} $

Answer

$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n} \frac{1}{1+(i / n)^{2}}=\int_{0}^{1} \frac{1}{1+x^{2}} d x$

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Video Transcript

some as indefinite Any girl. The first part is one over N that's going to be basically the width of each of the rectangle, so we could go ahead and convert that to DX. Um, the next thing we might notice is that we're only going from zero upto one. And the reason we know that is because it all rests on the definition of Delta X here. So let me explain that a little bit. It's b minus a over and and because that's all equal to one of rent, then we can basically assume that be is one and a zero. So that's how we end up with zero toe one there aan den. The last thing is that this expression times I will give us the of component here, and that just becomes X. So really the integration and is 1/1 plus X squared. And that would be the rewritten form of the limit on the some from above. There