Analysis Help: Lim SnTn is +Inf

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In summary, this conversation is discussing the theorem that states if two sequences (Sn) and (Tn) have limits of positive infinity and a positive number respectively, then the product of these sequences also has a limit of positive infinity. The conversation includes hints for the proof of this theorem and a reference to the source of the theorem and its proof. One person also asks for clarification on the question and shares their familiarity with the topic.
  • #1
Zygotic Embryo
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Let (Sn) and (Tn) be sequences such that the lim Sn = +inf and lim Tn > 0

Then lim SnTn = + inf
 
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  • #2
What is the question? And what work have you done on this problem?
 
  • #3
A friend of mine, gave it to me.

I don't know where to start
 
  • #4
its a statement

i need a proof
 
  • #5
what doyou know about sequences, and multiplication (or division) of sequences?
 
  • #6
is it

Let M > 0

Select a real number m so that 0 < m < limTn.

There exists an N1 such that:
n>N1 implies Tn>m

Since limSn=+inf there exists an N2 such that
n>N2 implies Sn>(M/m)

Set N = max{N1,N2}.

Then n>N imples SnTn>(M/m)*m = M
 
  • #7
I thought I recognized the wording you used for the statement of this theorem and its proof and sure enough, it is taken word for word out of the book "Elementary Analysis: the Theory of Calculus" by Kenneth Ross (pg 50-51). Its stated with proof as theorem 9.9

If you have the proof in front of you, why are you asking if that's it? Is there some part of the proof you don't understand?
 

1. What does "Lim SnTn is +Inf" mean?

Lim SnTn is a mathematical notation, where "Lim" stands for limit, "S" stands for the summation symbol, and "n" represents the number of terms in the sequence. The notation "+Inf" indicates that the limit of the sequence as n approaches infinity is positive infinity.

2. How is the limit of a sequence, with positive infinity, represented graphically?

The limit of a sequence with positive infinity is represented by a horizontal line on the graph, passing through the y-axis at a value of positive infinity. This indicates that as the number of terms in the sequence increases, the values of the sequence also increase and approach positive infinity.

3. What is the significance of the notation "+Inf" in "Lim SnTn is +Inf"?

The notation "+Inf" in "Lim SnTn is +Inf" indicates that the limit of the sequence is unbounded, meaning that the values of the sequence increase without bound as the number of terms increases. This is also known as an infinite limit.

4. How is the limit of a sequence, with positive infinity, calculated?

The limit of a sequence with positive infinity is calculated by finding the limit of the sum of the terms in the sequence. This involves taking the sum of the first n terms in the sequence and then taking the limit as n approaches infinity.

5. What are some real-life applications of "Lim SnTn is +Inf"?

The notation "Lim SnTn is +Inf" has applications in various fields such as physics, economics, and engineering. For example, in physics, it can be used to represent the infinite sum of forces acting on an object, and in economics, it can be used to represent the infinite growth of a company's profits over time. Additionally, in engineering, it can be used to represent the infinite limit of a sequence of values, such as the increase in the strength of a material as the number of layers increases.

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