Real Analysis, Sequence/series/supremum/infimum

[rayred]

Homework Statement

a) Given the definition of the divergence of a sequence {a_n} :
"For any H >0 we can find a number NH such that a_n >H, for all n>N_H"
prove that {a_n * b} diverges if {a_n } diverges for any b ≠ 0 .


b) Find the supremum and infimum for the se… 1 - 1/n } and, if possible, the maximum and minimum values. Explain your answers.

C) Consider the series
1/2 + 1/3^2 + 1/2^3 + 1/3^4 + 1/2^5 + ... = SUM from n = 1 to infiniti of a_n
where a_n = 1/2^n when n = odd ::: and a_n = 1/3^n when n = even.
By considering two subsequences of partial sums for odd and even n, show that the ratio test gives contradictory results.
The root test considers R = lim n→∞ | a_n |^1/n and states that when R < 1 the series
converges absolutely, R > 1 the series diverges and R = 1 the test gives no information. Use the root test to show that the series above converges.

 

[lippyka]

Homework Equations N/AThe Attempt at a Solution A) Given the definition of divergence, we can say that for any sequence {a_n}, if we take an H > 0, then there exists a number NH such that a_n > H, for all n > NH. Since b ≠ 0, we can multiply both sides by b to get b*a_n > bH, for all n > NH. Then, since bH is still greater than 0, if we take H' equal to bH, then we again have that b*a_n > H', for all n > N_(H'). Thus, since H' > 0, there must exist an N_(H') such that b*a_n > H', for all n > NH'. Therefore, we can conclude that {a_n*b} diverges if {a_n} diverges for any b ≠ 0. B) The supremum and infimum for the set {1 - 1/n} are the maximum and minimum values of the set, which are 1 and -infinity, respectively. The maximum value of 1 results when n = 1, while the minimum value of -infinity is the limit of 1 - 1/n as n→∞. C) First, when we consider the subsequences of partial sums for odd and even n, we see that the ratio test gives contradictory results because the ratio of the limit of the odd subsequence over the limit of the even subsequence is 1/2, while the limit of the odd subsequence over the limit of the even subsequence is 2. Thus, the ratio test does not give a conclusive result. When we use the root test, we find that R = lim n→∞ | a_n |^1/n = lim n→∞ (1/2^n)^1/n = 1/2. Since 1/2 < 1, this implies that the series converges absolutely.