The Real And Complex Number Systems

So, L 1 5 since L ≥ 1.
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Remark: (1) The sequence b n is the famous sequence named Fabonacci sequence.
There are many researches around it. Also, it is related with so called Golden Section,
5 −1
0.618....
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(2) The reader can see the book, An Introduction To The Theory Of Numbers by G.
H. Hardy and E. M. Wright, Chapter X. Then it is clear by continued fractions.
(3) There is another proof. We write it as a reference.
Proof: (STUDY) Sinceb n2 b n b n1 , we may think
x n2 x n x n1 ,
and thus consider x 2 x 1. Say and are roots of x 2 x 1, with . Thenlet
F n n − n
− ,
we have
F n b n .
So,itiseasytoshowthatL 1 5 . We omit the details.
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Note: The reader should be noted that there are many methods to find the formula of
Fabonacci sequence like F n . For example, using the concept of Eigenvalues if we can
find a suitable matrix.
Series
8.15 Test for convergence (p and q denote fixed rela numbers).
(a) ∑
n1
n 3 e −n
Proof: ByRoot Test, wehave
So, the series converges.
(b) ∑
n2
log n p
lim n→
sup
n3 1/n
e 1/e 1.
n
Proof: We consider 2 cases: (i) p ≥ 0, and (ii) p 0.
For case (i), the series diverges since log n p does not converge to zero.
For case (ii), the series diverges by Cauchy Condensation Theorem (or Integral
Test.)
(c) ∑ n1
p n n p (p 0)
Proof: ByRoot Test, wehave
lim n→
sup
pn 1/n
n p.
p
So, as p 1, the series diverges, and as p 1, the series converges. For p 1, it is clear
that the series ∑ n diverges. Hence,
and
∑
n1
p n n p converges if p ∈ 0, 1