The Real And Complex Number Systems

So,
(b) n2
1 1
n 2 −1
Proof: Consider
we have
So,
n2
1 − 1
2 n − 2
1
2 ∑ .
n1 nn1
1 1
n 2 − 1 n 2
n 2 − 1
n
k2
n2
1 1
k 2 − 1
1 1
n 2 − 1
2 ∑ 2 −n
n1
2.
nn
n − 1n 1 ,
n
kk
k − 1k 1
k2
2 n
n 1
2 1 − 1
n
2 ∑
k1
2 ∑
n1
2.
n 1
1
kk 1 .
1
nn 1
8.42 Determine all real x for which the product
n1
cosx/2 n converges and find the
value of the product when it does converge.
Proof: Ifx ≠ m, wherem ∈ Z, thensin x
n
k1
cosx/2 k 2n sin x
2 n
2 n sin x
n
2 n k1
2 n ≠ 0 for all n ∈ N. Hence,
cosx/2 k
sin x
2 n sin x → sin x x .
2 n
If x m, wherem ∈ Z. Then as m 0, it is clear that the product converges to 1. So,
we consider m ≠ 0 as follows. Since x m, choosing n large enough, i.e., as n ≥ N so
that sin x ≠ 0. Hence,
2 n
and note that
Hence,
n
k1
N−1
cosx/2 k
k1
lim n→
n
cosx/2 k cosx/2 k
kN
N−1
cosx/2 k sinx/2
N−1
2 n−N1 sinx/2 n
k1
sinx/2 N−1
2 n−N1 sinx/2 n sinx/2N−1
x/2 N−1 .
cosx/2 k sinx/2N−1 N−1
cosx/2
x/2 N−1 k .
k1
k1