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 one year ago
how do i find the limit as an goes to infinity of the following? (the equation's in the comments)
 one year ago
how do i find the limit as an goes to infinity of the following? (the equation's in the comments)

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alffer1
 one year ago
Best ResponseYou've already chosen the best response.0\[\frac{ 4 }{ n }\sum_{a=1}^{n}((\frac{ 4a }{ n })^{2} 4(\frac{ 4a }{ n }))\] This is the equation in question

alffer1
 one year ago
Best ResponseYou've already chosen the best response.0\[\frac{ 4 }{ n }\sum_{a=1}^{n}((\frac{ 4a }{ n })^{2})  \frac{ 4 }{ n }\sum_{a=1}^{n}(4(\frac{ 4a }{ n}))\]

alffer1
 one year ago
Best ResponseYou've already chosen the best response.0we're not sure where to go from here

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2looks like you are trying to compute some sort or reimann sum

alffer1
 one year ago
Best ResponseYou've already chosen the best response.0exactly. a right riemann sum of x^2 4x to be exact.

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2don't let the \(n\) through you off in the summation you are summing over \(a\) (never seen an \(a\) used like that before, but no matter) pull it right out front of the sum then use the formula for \[\sum_{a=1}^na^2\] and also \[\sum_{a=1}^na\]

tkhunny
 one year ago
Best ResponseYou've already chosen the best response.0After a little algebra. \(\dfrac{16}{n^{2}}\left[\dfrac{1}{n}\sum\limits_{a=1}^{n}a^{2} + \sum\limits_{a=1}^{n}a\right]\)

primeralph
 one year ago
Best ResponseYou've already chosen the best response.0@satellite73 Got this?

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2\[\frac{ 4 }{ n }\sum_{a=1}^{n}((\frac{ 4a }{ n })^{2})\] \[=\frac{64}{n^3}\sum_{a=1}^na^2\] for example @primeralph maybe, i am tired and all this latex is wearing me out

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2@tkhunny i think the \(4\) is being squared as well

tkhunny
 one year ago
Best ResponseYou've already chosen the best response.0Whoops. Make that 64, instead of 16.

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2not really sure the sum is correct, but assuming it is for the moment \[\frac{64}{n^3}\sum_{k=1}^nk^2=\frac{64}{n^3}\frac{n(n+1)(2n+1)}{6}\] for the first one

alffer1
 one year ago
Best ResponseYou've already chosen the best response.0Wait but how does this help us get the limit as n > infinity?

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2taking the limit as \(n\to \infty\) is very easy because the numerator is a polynomial of degree 3 will leading coefficient 128 and the denominator is a polynomial of degree 3 with leading coefficent 6

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2as you remember from some pre calc course about "horizontal asymptotes" that makes the limit \(\frac{128}{6}\)

alffer1
 one year ago
Best ResponseYou've already chosen the best response.0oh oops duh that makes sense.

satellite73
 one year ago
Best ResponseYou've already chosen the best response.2easy if you think about it that way instead of the way they do it in the text by dividing top and bottom by \(n^3\) and all that other nonsense

aef321
 one year ago
Best ResponseYou've already chosen the best response.0Thanks so much!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

alffer1
 one year ago
Best ResponseYou've already chosen the best response.0@satellite73 where did 128 come from?

alffer1
 one year ago
Best ResponseYou've already chosen the best response.0sorry probably a dumb question but...

alffer1
 one year ago
Best ResponseYou've already chosen the best response.0oh waitn.....nevermind
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