# Converting from a sum to an integral

#### ksdhart2

##### Senior Member
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

##### Elite Member
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
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
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)