Riemann sums

Probably been a bit neater to note its symmetric about pi/2 and using a centered form of the sum / formula like

But not worked it through.

Is there anything wrong in the working? Since I end up with root 3/4, and what is the formula that you mentioned? it doesn’t seem to show up in your reply

Your working is stilll a bit blurred. The ans is 1 as you say, so you must be dropping a simple factor somewhere. The formula is in the youtube (about 12 min). Its just a way to centreing the series which makes it easier to map the num and denom to trig terms.

Alright I guess there must be a mistake in there somewhere, may I ask is the way I evaluate the fraction near the end for N->infinity alright? For the top part I say it converges to 1 - e^pii/3 and for the bottom I try to use Taylor series ( I can’t just say 1 - e^0 )

Also how on earth do you do part b? Is there something I can look up because I’ve not actually seen a question like that before

Can you upload a better pic?

Of the question or the working?

Of the question or the working?

The pictures are quite clear it is just TSR puts them at a lower quality for some reason

But for part b what is the strategy?

Working but the question aint great either. For b) you should have something like
https://math.stackexchange.com/questions/139183/differentiating-definite-integral

But I’m this question x is also inside the function within the integral, how might I deal with this?

Do one thing at a time, can you get a better pic of your workinug uploaded somewhere?

Do one thing at a time, can you get a better pic of your workinug uploaded somewhere?

Looking back at your working, I cant help thinking your making it difficult for yourself with notation. Using
Nh = pi/3
and working it through with h and subbing at the end would simplfiy things. In particular, a few simple lines got me to something like
h [ e^(ipi/3) - e^(i2pi/3)] / [1 - e^ih]
and a bit of simple reasoning like that video seems to finish it off and take the imaginary part. There may be the odd typo in that.

Id guess youre error is coming when you pull e^ipi/3 out the Im(). Id guess your argument is that you take the real part so real*im, but youd also have an im*real part which seems to be missed out.

I’m not entirely sure what you mean, are we summing over h?

And twa that makes sense, it’s probably where the error is coming from

h is the usual width so the series is
h * [e^(ipi/3) e^(i(pi/3 h)) e^(i(pi/3 2h)) ... e^(i(pi/3 (N-1)h))
so sum over an index n going from 0 to N-1, but obviously N->inf => h->0, which is the usual argument to make at the end.

Ah okay, and you also suggest cantering it around pi/2 what exactly would that look like?