# Constructing a pair of sequences such that

• quarkycharmed
In summary: XIn summary, when given the task of constructing two sequences {a_n} and {b_n} that are both greater than or equal to zero for all integers n, decreasing, and divergent, but have a convergent series when the minimum of each term is taken, one approach is to use variants of 1/n, such as a_n = 1/n and b_n = 1/n^2. Another approach is to use the alternating series test, such as a_n = (-1)^n/n and b_n = (-1)^n/n^2. Always remember to check the conditions given in the problem and explore different sequences to find the solution.
quarkycharmed

## Homework Statement

The question asks me to construct a pair of sequences {a_n} from n=1 to inf and {b_n} from n=1 to inf such that a_n and b_n are both greater than or equal to zero for all integers n, both sequences are decreasing, and both series of these sequences diverge, but also such that the series c_n from n=1 to inf converges, where c_n is defined as the min{a_n,b_n}.

none

## The Attempt at a Solution

I've played around with a lot of sequences whose series I know to be convergent, mainly variants of 1/n, but I cannot figure out how to make two sequences such that when choosing the smaller nth term from the two sequences I wind up with a convergent series. The problem I see is that every convergent decreasing series I think of requires each term to get smaller much faster than any decreasing divergent series I can think of. I'm sure there is something really obvious I'm missing, so I don't know if anyone can give a hint without giving it away, but is there another angle from which I should look at this?

Thanks!

Thank you for your post. It seems like you are on the right track with your approach of using variants of 1/n. One way to construct the sequences {a_n} and {b_n} is to start with a_n = 1/n and b_n = 1/n^2. Both of these sequences are decreasing and divergent, but when you take the minimum of each term, c_n = min{a_n, b_n} = 1/n, which is a convergent series.

Another approach you can take is to use the alternating series test. For example, you can let a_n = (-1)^n/n and b_n = (-1)^n/n^2. Both of these sequences are decreasing and divergent, but when you take the minimum, c_n = min{a_n, b_n} = (-1)^n/n^2, which is a convergent series.

I hope this helps! Keep exploring different sequences and you will eventually find the solution. Remember to always check your work and make sure the conditions given in the problem are satisfied.
Scientist

## 1. What is the purpose of constructing a pair of sequences?

The purpose of constructing a pair of sequences is to compare and analyze two sets of data in order to find patterns, relationships, and differences between them. This can help in understanding the underlying mechanisms and principles of a particular system or phenomenon.

## 2. How do you construct a pair of sequences?

To construct a pair of sequences, you first need to identify the variables or elements that you want to compare. Then, you can collect data for each variable and organize it into a list or table. Finally, you can plot the data on a graph or perform statistical analysis to visualize and interpret the relationship between the two sequences.

## 3. What are the different types of sequences that can be constructed?

There are various types of sequences that can be constructed, such as arithmetic sequences, geometric sequences, and Fibonacci sequences. These sequences can have different patterns and relationships between the elements, which can be studied and analyzed using mathematical techniques.

## 4. Can a pair of sequences be used to make predictions?

Yes, a pair of sequences can be used to make predictions about future data points. By analyzing the patterns and trends in the two sequences, you can make educated guesses about what the next data points might be. However, it is important to note that these predictions may not always be accurate and can be affected by external factors.

## 5. What are some real-world applications of constructing a pair of sequences?

Constructing a pair of sequences has various real-world applications, such as in finance to analyze stock market trends, in biology to study genetic inheritance, and in physics to understand the behavior of physical systems. It can also be used in everyday life to track personal habits and behaviors, such as exercise routines or spending patterns.

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