   Chapter 11, 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 ) .

(a)

To determine

To prove: arccotxarccoty=arccotxy1+xy for xy1 .

Explanation

Result used:

The formula of tan(ab) is tanatanb1+tanatanb (1)

Proof:

Consider a=arccotx and b=arccoty , where 0<ab<π .

From equation (1),

cot(ab)=1tan(ab)=1+tanatanbtanatanb

That is,

cot(ab)=1+1cota1cotb1cota1cotb

cot(ab)=cotacotb+1cotbcota (2)

Substitute a and b values in equation (2),

cot(ab)=cot(arccotx<

(b)

To determine

To find: The sum of the series n=0arccot(n2+n+1) .

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