Solution of Problem 7 - Integral Test

By the :

f(x) = arctan(x)
x2+1
for x ³ 1 is continuous, positive and decreasing since

d
dx
 æ
ç
è 		arctan(x)
x2+1
 ö
÷
ø 		= 1-2(x)arctan(x)
( x2+1) 2
 £ 1-2(p/4)
( x2+1) 2
 < 0
   ( )

Since

ó
õ 		¥

1 
 arctan(x)
x2+1
 dx = ó
õ 		p/2

p/4 
 u du = 1
2
 æ
ç
è 		æ
ç
è 		p
2
 ö
÷
ø 		2

 
 - æ
ç
è 		p
4
 ö
÷
ø 		2

 
 ö
÷
ø
converges, then by the , the series

¥
å
 n = 1 
 | an| = ¥
å
 n = 1 
 arctan(n)
n2+1
also converges. Hence, our original series, åan,.

Note that this problem can also be solved using the Direct Comparison Test and the Limit Comparison Test.