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Material Type: Exam; Class: Live Music Production and Promotion; Subject: Music Industry; University: University of Southern California; Term: Fall 2009;
Typology: Exams
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Dave L. Renfro
n=
n=
n=
2 na 2 n is
often called the condensed series.^1 Note that we are not claiming the condensed series has the same sum as the original series.
series. Assume that 0 an+1 an for each n. We must prove that the biconditional X^1 n=
an < 1 ,
n=
2 na 2 n < 1 holds. This will be done by proving the two associated
conditional statements () and () separately.
n=
an < 1. Then
a 1 + a 2 + (a 3 + a 4 ) + (a 5 + a 6 + a 7 + a 8 ) + (a 9 + a 10 + + a 16 ) + < 1 =) a 1 + a 2 + (a 4 + a 4 ) + (a 8 + a 8 + a 8 + a 8 ) + (a 16 + a 16 + + a 16 ) + < 1 =) a 1 + a 2 + 2a 4 + 4a 8 + 8a 16 + < 1 : Doubling each term of this convergent series gives the following convergent series:
2 a 1 + 2a 2 + 4a 4 + 8a 8 + 16a 16 + < 1 :
Since the series above equals 2 a 1 +
n=
2 na 2 n , it follows that
n=
2 na 2 n < 1.
n=
an = 1. Then
a 1 + (a 2 + a 3 ) + (a 4 + a 5 + a 6 + a 7 ) + (a 8 + a 9 + + a 15 ) + = 1 =) a 1 + (a 2 + a 2 ) + (a 4 + a 4 + a 4 + a 4 ) + (a 8 + a 8 + + a 8 ) + = 1 =) a 1 + 2a 2 + 4a 4 + 8a 8 + = 1 :
Since the series above equals a 1 +
n=
2 na 2 n , it follows that
n=
2 na 2 n = 1.^2
(^1) More generally, the following holds. If 0 an+1 an for each n and fnk g is a subsequence of the positive integers such that M > 0 exists for which nk+1 nk M (nk nk 1 ) (i.e. the gaps between nk and nk+1 increase at most geometrically), then X^1 n=
an < 1 () X^1 k=
(nk+1 nk ) ank < 1 :
Putting fnk g^1 k=1 = 2 k^1 k=1 gives the Cauchy Condensation Test. (^2) What we have done is to prove P (^1) n=1 an = 1 ) P (^1) n=1 2 na 2 n = 1 , which is the contrapositive (under the assumption that 0 an+1 an) of P^1 n=1 2 na 2 n^ < 1 ) P^1 n=1 an < 1.
n=
n^2 The condensed series is
X^1 n=
2 n^
(2n)^2
n=
2 n
n=
n ;
which is a convergent geometric series.
n=
n The condensed series is X^1 n=
2 n^
(2n)
n=
which is clearly divergent.
n=
np^ The condensed series is
X^1 n=
2 n^
(2n)p
n=
2 n np
n=
21 p
n ;
which is a geometric series with r = 2^1 p. This diverges when jrj 1 , 21 p^ 1 , 1 p 0 , p 1 ; and converges when jrj < 1 , 21 p^ < 1 , 1 p < 0 , p > 1 :
n=
n(ln n)p^ The condensed series is
X^1
n=
2 n^
2 n^ (ln 2n)p
n=
(n ln 2)p
n=
np^ (ln 2)p
ln 2
p (^) X 1
n=
np^
which (use the previous example) converges when p > 1 and diverges when p 1.
n=
n ln n[ln(ln n)]p^ The condensed series is
X^1
n=
2 n^
2 n^ (ln 2n) [ln (ln 2n)]p
n=
(n ln 2) [ln (n ln 2)]p
n=
(ln 2) n [ln n + ln (ln 2)]p
which converges when p > 1 (direct comparison with the previous example) and diverges when p 1 (limit comparison with the previous example). (^3) In the examples that follow, observe how a use of the Cauchy Condensation Test transforms the series into one that converges (or diverges) more rapidly than the original series. In particular, the Cauchy Condensation Test transforms a p series into a geometric series (hence, these two types can be considered as relatives of each other) and it transforms an níth order logarithmic p series into an (n 1)íst order logarithmic p series.