How do you find this ? lim_(x->0)sum_(n=1)^oo(cosnx)/((4n-3)(4n+1))

1 Answer
Jun 8, 2017

1/4

Explanation:

lim_(x->0)sum_(n=1)^oo(cosnx)/((4n-3)(4n+1))=sum_(n=1)^oo 1/((4n-3)(4n+1))

but

1/((4n-3)(4n+1)) = 1/4(1/(4n-3)-1/(4n+1))

and

sum_(n=1)^m1/(4n-3)-sum_(n=1)^m 1/(4n+1) = 1-1/(4m+1) so

lim_(x->0)sum_(n=1)^oo(cosnx)/((4n-3)(4n+1))=1/4