If #a_n# converges and #a_n >b_n# for all n, does #b_n# converge?

1 Answer
Jun 1, 2018

See explanation below

Explanation:

We assume that we are talking about sequences (although, for infinite series, the reasoning is the quite similar)

Assume that #{a_n}# converges. This is the same #lim_(ntooo)a_n=L# with #L# a finite number

For all #{b_n}# we know that #b_n < a_n#

If #{b_n}# is a growing sequence, then all terms are under the value #L# and #{b_n}# converges perhaps to #L#

If #{b_n}# is a decreasing or alternate sequence, we can´t assume the convergence. Think in this example

#a_n=3+1/n# and #b_n=(-1)^n#

In this case #b_n < a_n# for all #n#, but #lim_(ntooo)a_n=3# and
#lim_(ntooo)b_n# doesn`t exist