## anonymous one year ago I do not understand how to do this.

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1. anonymous

Write limn→∞∑k=1n(2+k∗(5/n))3∗5/n as a definite Integral

2. tkhunny

I can't understand the notation. $$\lim_{n\rightarrow \infty}ln(2+k(5/n))^{3}\cdot\dfrac{5}{n}$$?

3. anonymous

yes

4. anonymous

that is the notation

5. tkhunny

Not quite. What is the "k=" doing in there?

6. anonymous

im not sure thats what my problem says though. Think it stands for constant

7. tkhunny

'k' maybe, but '='?

8. anonymous

oh im sorry let me rewrite equation real quick

9. tkhunny

Maybe you can use the [Draw] feature?

10. anonymous

$\lim_{n \rightarrow \infty}\sum_{k=1}^{n}(2+k*5/n)^3 *5/n$

11. tkhunny

Well, that makes more sense!

12. tkhunny

Did you just learn, in that moment, just enough LaTeX to write that? Good work.

13. anonymous

Thank you. Now i am unsure of how to convert that to a definite integral

14. tkhunny

Let's remember the definition of a Riemann Integral. Roughly, you chop things up into smaller and smaller pieces and never stop doing this. This is EXACTLY the function of 'n' in the given expression. $$n \rightarrow \infty$$

15. anonymous

right

16. anonymous

still to not understand how to convert

17. tkhunny

Try a couple values for n and see how it goes.

18. anonymous

what to do you mean try n values?

19. tkhunny

Do n=1, then do n = 2. Look for anything familiar or consistent.

20. anonymous

5*(7k)^3?

21. tkhunny

The first thing I noticed was that $$5/n$$ is constant for a given value of n. This suggests an equivalent expression: $$\lim_{n\rightarrow\infty}\dfrac{5}{n}\sum_{k=1}^{n}\left[2+\dfrac{5k}{n}\right]^{3}$$

22. anonymous

yes i agree with that

23. anonymous

and the two is constant

24. anonymous

can you just give me in converted? I like working back from the conversion

25. phi

I think of the 5/n as an increment (e.g. dx ) the k* 5/n ranges from 5/n to 5 for very large n, 5/n is close to 0, so k*5/n represents a variable x going from 0 to 5

26. anonymous

i do not understand what that means

27. phi

It helps if you have the "big picture" of how integration is summation of lots of thin triangles.

28. phi

*rectangles

29. tkhunny

$$=\lim_{x\rightarrow\infty}\dfrac{5}{n}\sum_{k=1}^{n}f\left(\dfrac{5k}{n}\right)$$ $$=\lim_{x\rightarrow\infty}\dfrac{b-a}{n}\sum_{k=1}^{n}f(x^{*})$$ $$=\int_{0}^{5}f(x)\;dx = \int_{0}^{5}\left(2+5x\right)^{3}\;dx$$

30. phi

I think it's just (2+x)^3 inside the integral

31. anonymous

okay! Thank you

32. tkhunny

Right. Typo. Thanks.