Convergence of Positive Sequences: Limits and Sums from 1 to Infinity

In summary, if a_n and b_n are both positive for all natural numbers (N) and lim a_n/b_n = 0 as n goes to infinity, then the sum of a_n converges if and only if the sum of b_n converges from 1 to infinity. This is because if lim a_n/b_n = 0, then there exists an N in natural numbers (N) such that |a_n/b_n - 0| < 0 for all n >= N, meaning that 0 < a_n < 0. Therefore, the sum of a_n from 1 to infinity is also 0 and thus, convergent. However, if lim a_n/b_n = 1, then the
  • #1
Askhwhelp
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Suppose a_n > 0 and b_n > 0 for all n in natural number (N). Also, lim a_n/b_n = 0 as n goes to infinity. Then the sum of a_n converges if and only if the sum of b_n converges ...both from 1 to infinity.

My approach is that lim a_n/b_n = 0 means that there exists N in natural number (N) for which |a_n/b_n - 0| < 0 for all n >= N. Then 0 < a_n < 0. The sum of a_n from 1 to infinity is 0. So The sum of a_n from 1 to infinity is convergent.

Is this proof that easy or I miss something?
 
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  • #2
Hi Askhwhelp! :smile:
Askhwhelp said:
Suppose a_n > 0 and b_n > 0 for all n in natural number (N). Also, lim a_n/b_n = 0 as n goes to infinity. Then the sum of a_n converges if and only if the sum of b_n converges ...both from 1 to infinity.

"if and only if"? … that obviously isn't true

do you mean "if", or do you mean lim a_n/b_n = 1 ? :confused:
|a_n/b_n - 0| < 0

sorry, but that doesn't even make sense :redface:

check the wording of the question, and start again :smile:​
 

What is the definition of convergence in a positive sequence?

Convergence in a positive sequence refers to the property that as the terms in a sequence increase, the sequence approaches a specific value known as the limit. In other words, as the terms increase, the difference between the terms and the limit becomes smaller and smaller.

How is the limit of a positive sequence calculated?

The limit of a positive sequence can be calculated by taking the limit of the individual terms in the sequence as the number of terms approaches infinity. This is denoted by the symbol "lim" and is typically written as "lim (n->∞)".

What is the significance of the limit in a positive sequence?

The limit in a positive sequence is significant because it represents the ultimate value that the sequence is approaching. It can also help determine if a sequence is convergent or divergent, with a convergent sequence having a finite limit and a divergent sequence having no limit.

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

A convergent positive sequence is one where the terms get closer and closer to a specific limit as the number of terms increases. In contrast, a divergent positive sequence is one where the terms do not approach a specific limit and may increase or decrease without bound.

How is the sum of a positive sequence calculated?

The sum of a positive sequence can be calculated by adding all of the terms in the sequence together. This can be done for a finite number of terms or by taking the limit of the sum as the number of terms approaches infinity.

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