# Summing up this series

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.

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