Question

I do not understand how to do this.

Answer 1

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

Answer 2

I can't understand the notation.

\(\lim_{n\rightarrow \infty}ln(2+k(5/n))^{3}\cdot\dfrac{5}{n}\)?

Answer 3

yes

Answer 4

that is the notation

Answer 5

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

Answer 6

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

Answer 7

'k' maybe, but '='?

Answer 8

oh im sorry let me rewrite equation real quick

Answer 9

Maybe you can use the [Draw] feature?

Answer 10

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

Answer 11

Well, that makes more sense!

Answer 12

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

Answer 13

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

Answer 14

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. \(n \rightarrow \infty\)

Answer 15

right

Answer 16

still to not understand how to convert

Answer 17

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

Answer 18

what to do you mean try n values?

Answer 19

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

Answer 20

5*(7k)^3?

Answer 21

The first thing I noticed was that \(5/n\) is constant for a given value of n. This suggests an equivalent expression:

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

Answer 22

yes i agree with that

Answer 23

and the two is constant

Answer 24

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

Answer 25

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

Answer 26

i do not understand what that means

Answer 27

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

Answer 28

*rectangles

Answer 29

\(=\lim_{x\rightarrow\infty}\dfrac{5}{n}\sum_{k=1}^{n}f\left(\dfrac{5k}{n}\right)\)

\(=\lim_{x\rightarrow\infty}\dfrac{b-a}{n}\sum_{k=1}^{n}f(x^{*})\)

\(=\int_{0}^{5}f(x)\;dx = \int_{0}^{5}\left(2+5x\right)^{3}\;dx\)

Answer 30

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

Answer 31

okay! Thank you

Answer 32

Right. Typo. Thanks.