# Summing up this series

• utkarshakash
In summary, the conversation discusses the sums An and Bn for n=1,2,3... and presents four options to determine the relationship between these sums. It is mentioned that evaluating A(2010) directly is challenging and the formula arccosx + arccosy = arccos{xy − √[(1 − x2)(1 − y2)]} may not be helpful. Instead, the sums can be thought of as Riemann sums with a function of arccos(x) and the integral can be used to determine the relationship between the sums. The correct options cannot be determined solely by evaluating the sums directly.

Gold Member

## Homework Statement

Let An = $\sum_{r=1}^n arccos \frac{r}{n}$ and Bn=$\sum_{r=0}^{n-1} arccos \frac{r}{n}$ for n=1,2,3... then

A)A(2010) < 2010 and B(2010)>2010
B)A(2010) > 2010 and B(2010)<2010
C)A(n) < B(n) for all n
D)$\lim_{n \to \infty} \dfrac{A_n}{n} = 1$

More than one option is correct.

## The Attempt at a Solution

Finding A(2010) seems really challenging. I start by writing out all the terms manually.

arccos(1/2010)+arccos(2/2010)+arccos(3/2010)+arccos(4/2010)+... ..arccos(2010/2010)

Atmost I can only simplify the last term to 0. But what about the rest of the terms? The formula arccosx + arccosy = arccos{xy − √[(1 − x2)(1 − y2)]} doesn't help. It only makes the terms more complicated.

For A and B there is no way they expect you to evaluate that series directly - it's going to come from a more general statement like A(n) < n for all n, or A(n) > n for all n.

As far as how to do that I recommend thinking about these series as Riemann sums (modulo a little algebra) if you have covered that.

Office_Shredder said:
For A and B there is no way they expect you to evaluate that series directly - it's going to come from a more general statement like A(n) < n for all n, or A(n) > n for all n.

As far as how to do that I recommend thinking about these series as Riemann sums (modulo a little algebra) if you have covered that.

So this is basically an integral where subintervals are of unit length and the function is arccos(x) and that instead of finding the sum I should find the integral. Is this what you mean?