# Real Analysis, Sequence/series/supremum/infimum

## 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.