   Chapter 11.P, Problem 8P

Chapter
Section
Textbook Problem

(a) Prove a formula similar to the one in Problem 7(a) but involving arccot instead of arctan.(b) Find the sum of the series ∑ n = 0 ∞   arccot ( n 2 + n + 1 ) .

To determine

Part (a):

To prove:arccotx-arccoty=arccot1+xyy-x for y x

if the left side lies between  0<a-b<π

Explanation

1) Concept:

Use the formula of  tanx-y, and then simplify by rearranging the terms.

2) Formula:

tana-b=tana-tanb1+tana tanb

3) Calculation:

Let a=arccotx and b=arccoty where 0<a-b<π

Thus,

cota-b=1tana-b

By using the formula,

=1+tana tanbtana-tanb

It can be rearranged as

=1cota·1cotb+11cota-1cotb· cotacotbcotacotb

By simplifying,

=1+cotacotbcotb-cota

Substitute the values of a and b

To determine

Part (b):

To find:

The sum of the series

n=0arccotn2+n+1

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