I do not understand how to do this.

- anonymous

I do not understand how to do this.

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- anonymous

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

- tkhunny

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

- anonymous

yes

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## More answers

- anonymous

that is the notation

- tkhunny

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

- anonymous

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

- tkhunny

'k' maybe, but '='?

- anonymous

oh im sorry let me rewrite equation real quick

- tkhunny

Maybe you can use the [Draw] feature?

- anonymous

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

- tkhunny

Well, that makes more sense!

- tkhunny

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

- anonymous

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

- 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. \(n \rightarrow \infty\)

- anonymous

right

- anonymous

still to not understand how to convert

- tkhunny

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

- anonymous

what to do you mean try n values?

- tkhunny

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

- anonymous

5*(7k)^3?

- tkhunny

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

- anonymous

yes i agree with that

- anonymous

and the two is constant

- anonymous

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

- 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

- anonymous

i do not understand what that means

- phi

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

- phi

*rectangles

- tkhunny

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

- phi

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

- anonymous

okay! Thank you

- tkhunny

Right. Typo. Thanks.

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