Real Analysis: Product of sequences diverges

In summary, if a sequence a_n diverges to positive infinity and a sequence b_n converges to 0, we can prove that their product a_n*b_n also diverges to positive infinity. This can be proven by choosing appropriate values for N based on the behavior of the two sequences.
  • #1
tarheelborn
123
0

Homework Statement



If a_n diverges to +inf, b_n converge to 0; prove a_n*b_n diverges to +inf

Homework Equations





The Attempt at a Solution



My attempt follows: I seem to have trouble getting things in the right order, so I am trying to work on my technique, with your help. Also, I am afraid I may have omitted reference to some theorem that I am taking for granted, which is another of my bad habits. Please review for me and advise as appropriate. I am determined to conquer this subject! Thanks.

Let M, e > 0, M, e \in R. By definition of a limit of a sequence, we can choose N_b such that |b_n - M|< e, n >= N. Then -e < b_n - M < e, so M - e < b_n < M + e. So b_n > M - e. We can then choose N_a such that a_n >= M/(M-e), n >= N. Let N = max (N_b, N_a). Then a_n*b_n >= M/(M-e), n >= N. Thus, {a_n*b_n} diverges to + infinity.
 
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  • #2
Counter example: [tex] a_n = n \left , \left b_n = \frac{1}{n^2} [/tex]
 
  • #3
Sorry... b_n converges to M > 0.
 
  • #4
so you need to show that for any P > 0 you can find N such that anbn > P forall n > N

bn converges to M, so for any e > 0, there exists N1 such that |bn-M|< e for all n>N1

an diverges to infinity, so for any P1 > 0, there exists N2 such that an > P1 for all n > N2

so think about how to pick N based on the behaviour of bn & an
 
  • #5
Actually, I did get this solved. Thank you so much for your help!
 

1. What is a product of sequences in real analysis?

In real analysis, a product of sequences refers to the multiplication of two sequences of real numbers term by term. This is similar to the concept of a product of two functions in calculus, where the output of the product is the product of the individual outputs of the two functions.

2. How do you determine if a product of sequences diverges?

To determine if a product of sequences diverges, you can use the following theorem: If one of the sequences diverges to infinity and the other sequence converges to a non-zero limit, then the product of the two sequences diverges. Additionally, if both sequences converge to zero, the product may still converge or diverge depending on the rate of convergence.

3. What is the difference between a divergent product and a convergent product?

A divergent product of sequences means that the product does not have a finite limit as n goes to infinity, while a convergent product means that the product has a finite limit as n goes to infinity. In other words, a divergent product can be thought of as going to infinity, while a convergent product can be thought of as approaching a specific value.

4. Can a product of sequences converge to a non-zero limit?

Yes, a product of sequences can converge to a non-zero limit. This can happen if one of the sequences converges to a non-zero limit and the other sequence also converges to a non-zero limit, or if one of the sequences converges to a non-zero limit and the other sequence diverges to infinity at a slower rate.

5. Can a product of sequences converge to zero?

Yes, a product of sequences can converge to zero. This can happen if one of the sequences converges to zero and the other sequence converges to a non-zero limit, or if both sequences converge to zero at a faster rate than the product diverges to infinity.

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