Does \( a_{n} \) Diverge to \(\infty\)?

In summary, using the given conditions of \left\{b_{n}\right\}\rightarrow\infty and \frac{a_{n}}{b_{n}}\rightarrow C as n\rightarrow\infty, it can be proven that \left\{a_{n}\right\}\rightarrow\infty as n\rightarrow\infty. This is done by showing that \left\{a_{n}\right\} cannot be bounded and cannot diverge to -\infty, and then using the epsilon definition to choose the "right" value and directly prove the desired result.
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Homework Statement



Given that [tex]b_{n}\rightarrow\infty[/tex] and [tex]\frac{a_{n}}{b_{n}}\rightarrow C[/tex] (where C>0) as [tex]n\rightarrow\infty[/tex], prove that [tex]a_{n}[/tex] must also diverge to [tex]\infty[/tex], that is, [tex]a_{n}\rightarrow\infty[/tex] as [tex]n\rightarrow\infty[/tex]

Homework Equations



As above.

The Attempt at a Solution



I could easily deduce that [tex]\left\{a_{n}\right\}[/tex] cannot be bounded, otherwise the sequence of quotients will be null. Also, I deduced that [tex]\left\{a_{n}\right\}[/tex] cannot diverge to [tex]-\infty[/tex] as the limit of convergence cannot be positive in this case. However, I could not deduce that [tex]\left\{a_{n}\right\}[/tex] must be unbounded BOTH above and below, and so I cannot rule out the possibility of oscillation, yet.

Originally I said if [tex]\left\{a_{n}\right\}[/tex] oscillates AND also unbounded, then it is always possible to find a negative term no matter what the value of n is. Since [tex]\left\{b_{n}\right\}[/tex] is always positive for large n, then it is always possible to find the term of [tex]\left\{\frac{a_{n}}{b_{n}}\right\}[/tex] that is negative, therefore cannot converge to a positive limit. However it has become obvious that "unbounded" doesn't mean unbounded in both ways, i.e. sequences like {1,2,1,4,1,8,1,16,...} are unbounded and oscillating but will not give a negative term, so my argument isn't valid.

But apparently the easier approach is to use the epsilon definition, chosen the "right" epsilon and get the result directly without having to resort to proof by cases. I can't figure out what the "right" value is.
 
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Don't worry, I've got it (after thinking about it for a long time) by setting epsilon=l/2
 

FAQ: Does \( a_{n} \) Diverge to \(\infty\)?

What is a sequence in real analysis?

A sequence in real analysis is a list of numbers arranged in a specific order. Each number in the sequence is called a term, and the position of the term in the sequence is called its index. Sequences are often denoted using the notation {an}, where n represents the index of the term.

What is the difference between a convergent and a divergent sequence?

A convergent sequence is one in which the terms approach a specific limit as the index increases. In other words, the terms get closer and closer to a particular value. On the other hand, a divergent sequence is one in which the terms do not approach a specific limit and instead either increase or decrease without bound.

How is the limit of a sequence determined?

The limit of a sequence is determined by looking at the behavior of the terms as the index approaches infinity. If the terms approach a specific value, that value is the limit of the sequence. If the terms do not approach a specific value, the sequence is said to not have a limit.

What is the Cauchy criterion in real analysis?

The Cauchy criterion is a method for determining whether a sequence is convergent. According to this criterion, a sequence is convergent if and only if, for any positive number ε, there exists a positive integer N such that for all indices m and n greater than or equal to N, the difference between the mth and nth terms of the sequence is less than ε.

How is the Bolzano-Weierstrass theorem used in real analysis?

The Bolzano-Weierstrass theorem states that any bounded sequence in real analysis has a convergent subsequence. This theorem is useful in proving the convergence of a sequence by showing that it is bounded and then finding a convergent subsequence within it.

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