How do I express a limit to infinity as a definite integral? (Example Inside)

A homework question? At least you’re being obvious about it.
Sorry, I don’t help others with homework. I never got help with mine, you see, and I’m bitter.

@dynamicduo, I’m looking to learn this concept. I’ve got 20 other questions just like it. Notice how I didn’t just post “oh do all 20 of these for me please so I can copy them and get an A on this set.” Unlike lots of kids that just want to copy homework, I realize that the test cometh and that I’m going to have to learn it eventually. Thus, I don’t want an answer, I want an explanation.

I’m confused…you have to different variables going on.
You wrote this, using an x in the limit definition but an n in the expression. If you could fix or explain whatever is going on there, I’ll try to help you out.

@lefteh, Sorry about that. Should be limit as n approaches infinity. It’s n because that’s the variable in a summation or w/e. I’m just too used to writing x :/

As this seems to be homework I am just going to help you do that. I am not going to solve it for you.

You need to write that as a Riemann sum for some function. Look at the general term of your sum. After multiplying 1/n through we get it in the following form:

(1/n) * (i/n)^3

What is a Riemann sum of a function f(x)? It is the sum of terms in form f(xi*) * (xi – x(i-1)) Where xi* is a sample point in the i-th interval [ x(i-1), xi ].

Here is a hint for your example: (xi – x(i-1)) in your example is 1/n. Figure out the function f(x) that its Riemann sum would give what you are looking for, by looking at (i/n)^3.

What interval has been divided into n small subintervals of equal length? That would give you the lower and upper limits of your definite integral.

Let me know if you need more help. Good luck!

^ Great answer. Couldn’t have put it any better myself.

@BonusQuestion Thank you for your help! I ended up figuring all the questions out, even the challenge one. You’re a calc lifesaver!

By definition the index of your sum has to be positive, thus:

[infinity]
[sigma] s(n)
[k=1]

= lim [n->(infinity)] sum above
= lim [k-> b] [integral] s(k) | k = 1 to k = b
= lim [k-> b] S(1) – lim [k->b] S(b)
= lim [k-> b] S(1) – lim [k-> (infinity)] S(infinity)

That’s a crude representation, I hope it helps.

As an additional bit of info, if when you apply the FToC and the limits, and get infinity, then the series diverges. If you get anything else, it either converges or diverges.