Question

Determine if the following statements are true or false. If true give a reason or cite a theorem and if false, give a counter example.
a. If {a _n} is bounded, then it converges.
b. If {a _n} is not bounded, then it diverges.
c. If {a _n} diverges, then it is not bounded

Give an example of divergent sequences {a _n} and {b _n} such that {a _n + b _n} converges

Answer 1

Answer:

a. False the series $\sum\limits_{i=1}^\infty (-1)^n$is bounded but does not converge

b. False, xₙ = n + (-1)ⁿ⁻¹(n - 1) does not diverge to infinity but it is not bounded for n ≥ 1

c. False some bounded sequences are divergent

An example of divergent sequences, aₙ, and bₙ, such that aₙ + bₙ converges is $a_n = \sum\limits_{n} \dfrac{1}{n} , \, b_n = \sum\limits_{n} \dfrac{-1}{n}$

$\sum\limits_{n} \dfrac{1}{n} + \sum\limits_{n} \dfrac{-1}{n}$ is convergent

Step-by-step explanation: