- #1
Rectifier
Gold Member
- 313
- 4
The problem
I want to calculate $$\sum^n_{k=1} \frac{4}{1+ \left(\frac{k}{n} \right)^2} \cdot \frac{1}{n}$$ when ##n \rightarrow \infty##
The attempt
## \sum^n_{k=1} \underbrace{f(\epsilon)}_{height} \underbrace{(x_k-x_{k-1})}_{width} \rightarrow \int^b_a f(x) \ dx ##, when ##n \rightarrow \infty## ##n=##amount of rectangles and ##e## is the height of a single rectangle on a specific sub-interval ##k##.
I set ##f(x)=\frac{4}{1+ \left(x \right)^2}## and then for some reason the suggestion in my book is to set ##x_k = \frac{k}{n}## but I don't understand why.
I need help with calculating the end-points for the integral ##a## and ##b## later too.
Pleas help.
I want to calculate $$\sum^n_{k=1} \frac{4}{1+ \left(\frac{k}{n} \right)^2} \cdot \frac{1}{n}$$ when ##n \rightarrow \infty##
The attempt
## \sum^n_{k=1} \underbrace{f(\epsilon)}_{height} \underbrace{(x_k-x_{k-1})}_{width} \rightarrow \int^b_a f(x) \ dx ##, when ##n \rightarrow \infty## ##n=##amount of rectangles and ##e## is the height of a single rectangle on a specific sub-interval ##k##.
I set ##f(x)=\frac{4}{1+ \left(x \right)^2}## and then for some reason the suggestion in my book is to set ##x_k = \frac{k}{n}## but I don't understand why.
I need help with calculating the end-points for the integral ##a## and ##b## later too.
Pleas help.