Sigmatic Trig Function help

Question

wwb00 · Answer

$$\lim_{n \rightarrow \infty}(\sum_{\theta=0}^{n}(\sec(\theta*(\pi/4n))/(n+1))$$

wwb00 · Answer

I am calculating the average of the lengths of every line coming from one vertex of a square. This came from my calculations, and I can not simplify the sigma within the limit.

wwb00 · Answer

Does anyone have a plausible approach? I have graphed this and it converges but I want an exact solution.

wwb00 · Answer

|dw:1468175089443:dw|

wwb00 · Answer

I don't care about the other parts. Is there a way, any way at all, to do this Sigma?

amistre64 · Answer

the line goes from one vertex all the way to another side?

wwb00 · Answer

yes

amistre64 · Answer

do you have to use the trig in it?  or was that your own idea?

wwb00 · Answer

my own idea. Is there a better way ?

amistre64 · Answer

not sure if better is the right term ... but i was thinking of pythaging it.  

line^2 = side^2+height^2  as side does from 0 to side

wwb00 · Answer

yes so are you supposing a related rate problem?

amistre64 · Answer

something like that ...

wwb00 · Answer

what is it that you suppose? could we define x and y parametrically?

amistre64 · Answer

maybe, as x moves from 0 to 1  or whatever the side length is of the square

d=sqrt(1+x^2)

this gives us half the lines

$$\int d=\int^{1}_{0}\sqrt{1+x^2}~dx$$

as a start maybe?

amistre64 · Answer

|dw:1468176393708:dw|

times 2 gives us half the lines in the square right?