Question

Explain why or why notDetermine whether the following statements are true and give an explanation or counterexample.a. The terms of the sequence {aₙ} increase in magnitude, so the limit of the sequence does not exist.

Accepted Answer

First, understand what it means for the terms of a sequence $$ \{a_n\}  to $$increase in magnitude. This means that the absolute values $$ |a_n| $$ are getting larger as $$ n $$ increases. Recall that a sequence $$ \{a_n\} $$ converges to a limit $$ L  if $$and only if the terms get arbitrarily close to $$ L  as$$ $$ n 	o \infty . If $$the magnitude $$ |a_n| $$ increases without bound, the terms cannot approach a finite limit. However, consider the possibility that the terms might oscillate or approach zero despite increasing magnitude. For example, if $$ a_n = (-1)^n n $$, the magnitude increases but the sequence does not converge because it oscillates and grows without bound. On the other hand, if the magnitude increases but the terms approach zero, this would be a contradiction because the magnitude cannot increase and approach zero simultaneously. Therefore, if the terms of the sequence increase in magnitude without bound, the sequence does not have a finite limit. This means the statement is true: increasing magnitude implies the limit does not exist.

Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
a.The terms of the sequence {aₙ} increase in magnitude, so the limit of the sequence does not exist.

Verified video answer for a similar problem:

Key Concepts

Sequence and Limit of a Sequence

Magnitude (Absolute Value) of Sequence Terms

Convergence Criteria and Counterexamples

Explain why or why notDetermine whether the following statements are true and give an explanation or counterexample.a.The terms of the sequence {aₙ} increase in magnitude, so the limit of the sequence does not exist.