Proving Divergence of (tn + sn)

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In summary, the conversation revolved around finding a proof for the statement that if (tn) is divergent and (sn) is convergent, then (tn+sn) is divergent. The speaker mentioned a complex approach involving considering unbounded and oscillatory sequences, as well as convergent subsequences, but was unsure about the validity of their conclusion. They were hoping for a simpler contradiction proof using the triangle inequality.
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dmac1215
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I have a super round about way to prove this, but I'm having trouble finding a succinct proof

Let (tn) be diverge and (sn) converge. Show (tn+sn) diverges

The way I was doing involved considering that tn was unbounded, then showing it (sn+tn) is divergent.

Then I had to consider that tn is bounded and oscillatory, consider convergent subsequences, and show (sn+tn) had no unique limit, and therefore diverges. This part of the proof seemed less clear and I'm not sure if I can assert that because I have convergent subsequences of (tn) with multiple limits that (sn+tn) also has multiple limits.

I figure there has got to be some simple contradiction proof involving some triangle inequality trick that I'm just missing.

Thanks
 
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  • #2
Hint: the sum (or difference) of two convergent sequences is convergent.
 
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1. What is the definition of "Proving Divergence of (tn + sn)"?

The term "Proving Divergence of (tn + sn)" refers to the process of showing that the series tn + sn diverges, meaning that it does not have a finite sum. This involves using mathematical techniques and principles to demonstrate that the terms of the series do not approach a specific value as n approaches infinity.

2. Why is proving the divergence of (tn + sn) important in mathematics?

Proving the divergence of (tn + sn) is important because it allows us to understand the behavior of a series and determine whether it has an infinite or finite sum. This information is crucial in many mathematical applications, such as in the study of limits, integrals, and differential equations.

3. What are some common techniques used to prove the divergence of (tn + sn)?

Some common techniques used to prove the divergence of (tn + sn) include the comparison test, the limit comparison test, and the integral test. These methods involve comparing the series in question to other known divergent series or using integrals to evaluate the convergence of the series.

4. Can a series diverge even if its individual terms approach zero?

Yes, a series can still diverge even if its individual terms approach zero. This is because the convergence or divergence of a series is determined by the behavior of the terms as n approaches infinity, not their individual values. A series can have infinitely many small terms, but if they do not approach a specific value, the series will still diverge.

5. How does proving the divergence of (tn + sn) relate to the concept of divergence in calculus?

In calculus, the concept of divergence refers to the behavior of a vector field as it approaches a point. Similarly, proving the divergence of (tn + sn) involves analyzing the behavior of a series as its terms approach infinity. Both concepts involve the idea of approaching a limit, but in different contexts.

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