Converting from a sum to an integral

ksdhart2

Senior Member
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Mar 25, 2016
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Hi all. I've received a problem from a student I'm tutoring, and I'm having some difficulties figuring it out. Here's the full and exact problem text:

Evaluate the following limit as a definite integral (Hint: Consider \(f(x) = x^4\))

\(\displaystyle \lim_{n \to \infty} \sum\limits_{k=1}^{n} \frac{k^4}{n^5}\)
I think I'm meant to "see" a Riemann Sum in there somewhere, but I just don't. In order to put in the form of a right Riemann Sum:

\(\displaystyle \sum\limits_{k=1}^{n} f(x_i) \Delta x\) where \(\displaystyle \Delta x = \frac{b - a}{n}\)

Wouldn't I need to know a value for \(a\) and \(b\)? It doesn't seem like those were given anywhere... or am I supposed to intuit it from the problem somehow? And how can I deal with the \(n^5\) term?

I feel like I'm missing something obvious, and I'll probably be ashamed once you guys point it out to me, but for now I'm completely at a loss. Any help would be greatly appreciated.
 

Dr.Peterson

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Nov 12, 2017
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I see the summand as (k/n)^4 (1/n), where 1/n would be \(\displaystyle \Delta x\) and k/n would be x. From that, you can determine what a and b have to be.
 

ksdhart2

Senior Member
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Mar 25, 2016
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1,296
Ah. I suspected it'd be something that just wasn't clicking. Thanks! I can solve for \(a\) and \(b\) because I know \(b - a = 1 \implies b = a + 1\) and then I need \(x_k = \frac{k}{n}\), so:

\(\displaystyle x_k = a + k \Delta x = a + \frac{k}{n}\)

\(\displaystyle x_k = \frac{k}{n} \implies a = 0 \implies b = 1\)

Way easier than I was making it out to be when I was spinning my wheels earlier.
 

Jomo

Elite Member
Joined
Dec 30, 2014
Messages
4,049
Hi all. I've received a problem from a student I'm tutoring, and I'm having some difficulties figuring it out. Here's the full and exact problem text:



I think I'm meant to "see" a Riemann Sum in there somewhere, but I just don't. In order to put in the form of a right Riemann Sum:

\(\displaystyle \sum\limits_{k=1}^{n} f(x_i) \Delta x\) where \(\displaystyle \Delta x = \frac{b - a}{n}\)

Wouldn't I need to know a value for \(a\) and \(b\)? It doesn't seem like those were given anywhere... or am I supposed to intuit it from the problem somehow? And how can I deal with the \(n^5\) term?

I feel like I'm missing something obvious, and I'll probably be ashamed once you guys point it out to me, but for now I'm completely at a loss. Any help would be greatly appreciated.
There is a very minor typo, it should be f(xk) and not f(xi)
 
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