About Convergence of a particular sequence

In summary, the conversation discusses defining an infinite sequence {b_n} and proving that {(b_n)/n} is a convergent sequence. The homework statement defines the sequence and the homework equations provide information about the sequence. The attempt at a solution involves using induction to show that 0 <= b_n / n <= b_1, but the individual is stuck on how to show that it is monotone. A suggestion is given to show that b_1/n^2 is a convergent sequence using the integral test.
  • #1
evalover1987
15
0

Homework Statement

let's define {b_n} as an infinite sequence such that
all terms are nonnegative, and
for any i, j >= 1,

b_{i+j} <= b_i + b_j

it's easy to show that b_n/n <= (b_1) for all n, but
how would you show that
{(b_n)/n} is a convergent sequence?

Homework Equations


The Attempt at a Solution


Homework Statement


Homework Equations


The Attempt at a Solution



Using induction, I got upto 0 < = b_n / n < = b_1
but that was pretty much it.
I have to show that it's monotone from some N on or something like that, i think..
but I'm just stuck about how to show that.
 
Last edited:
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  • #2
If you've shown b_n<b_1/n then b_n/n<b_1/n^2. Can you show b_1/n^2 is a convergent sequence? An integral test should work.
 
  • #3
my bad, problem fixed. i meant b_n < = b_1
 

1. What is the definition of convergence of a sequence?

Convergence of a sequence refers to the behavior of the terms in a sequence as the number of terms increases. A sequence is said to converge if its terms approach a specific value or limit as the number of terms increases.

2. How is convergence of a sequence different from divergence?

Convergence and divergence are opposite concepts. A sequence is said to diverge if its terms do not approach a specific value or limit as the number of terms increases. In other words, the terms of a divergent sequence become increasingly larger or smaller, without approaching a specific value.

3. What is the difference between convergent and oscillating sequences?

A convergent sequence approaches a specific value or limit as the number of terms increases, while an oscillating sequence does not have a specific value or limit, but its terms alternate between two or more values.

4. How do you determine the convergence of a particular sequence?

To determine the convergence of a sequence, you can use various methods such as the limit test, comparison test, or ratio test. These tests involve evaluating the behavior of the terms in the sequence and determining if they approach a specific value or limit.

5. Can a sequence be both convergent and divergent?

No, a sequence cannot be both convergent and divergent at the same time. A sequence can only have one of these two behaviors - either its terms approach a specific value or limit (convergent) or they do not (divergent).

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