Question

Write the definite integral for the summation: the limit as n goes to infinity of the summation from k equals 1 to n of the product of the square of the quantity 1 plus k over n squared and 1 over n.

Answer 1

https://gyazo.com/478f27e7e70fd5a27beb5d16043fe87e

Answer 2

As n approaches to infinity the values get closer to 0

Answer 3

Indeed, As n approaches to infinity the values get closer to 0

Answer 4

Sorry went to use the restroom.

Answer 5

And now that I take a good look what I was thinking would make no sense here.

Answer 6

Hint:


\[\Large \lim_{n \to \infty}\sum_{k = 1}^{n}\left(1+\frac{k}{n}\right)^2\left(\frac{1}{n}\right)\]


\[\Large \lim_{n \to \infty}\sum_{k = 1}^{n}\left(1+\frac{1}{n}*k\right)^2\left(\frac{1}{n}\right)\]


\[\Large \lim_{n \to \infty}\sum_{k = 1}^{n}\left({\color{red}{1}}+{\color{blue}{\frac{1}{n}}}*k\right)^2\left({\color{blue}{\frac{1}{n}}}\right)\]


\[\Large \lim_{n \to \infty}\sum_{k = 1}^{n}\left({\color{red}{a}}+{\color{blue}{\Delta x}}*k\right)^2\left({\color{blue}{\Delta x}}\right)\]

Answer 7

Keep in mind that
\[\Large x_k = a+\Delta x*k\]
\[\Large \Delta x = \frac{b-a}{n}\]

Answer 8

So b=2

Answer 9

yep

Answer 10

But what is Xk for?

Answer 11

xk represents a general term in the sequence x1,x2,x3, ..., xn

Answer 12

the idea is that you can find the approx area under the curve by breaking up the interval into n equal pieces and finding the area of each rectangular piece

Answer 13

this might help visualize an example

https://www.math.hmc.edu/calculus/tutorials/riemann_sums/gif/figure6.gif

Answer 14

But Xk would be equal to what's in the first parenthesis so how will that help me when writing the integral?

Answer 15

Integral is equal to area under curve

Answer 16

Hint: f(x) ----> f(xk)

Answer 17

What?

Answer 18

Do you see how I got\[\Large \lim_{n \to \infty}\sum_{k = 1}^{n}\left({\color{red}{1}}+{\color{blue}{\frac{1}{n}}}*k\right)^2\left({\color{blue}{\frac{1}{n}}}\right)\]

Answer 19

Yes I understand the concept just not how it will be applied

Answer 20

that will turn into

\[\Large \lim_{n \to \infty}\sum_{k = 1}^{n}\left(x_k\right)^2\left({\color{black}{\frac{1}{n}}}\right)\]

Answer 21

the \(\LARGE (x_k)^2\) portion represents the f(x) function

the remaining bit is the delta x

Answer 22

Ohhhhhhhh
I feel stupid now :/

Answer 23

Thanks

Answer 24

Don't feel stupid. You're still learning :)

Answer 25

we have n rectangles

each rectangle is delta x = 1/n units wide

the height of each rectangle is equal to f(x_k) = (1+k/n)^2 units

Answer 26

Got it Thanks