Zero Limit of Sum of Squares of Terms with Bounded Range

In summary, the concept of "Zero Limit of Sum of Squares of Terms with Bounded Range" refers to the situation where the sum of the squares of terms in a sequence approaches zero as the range of the terms becomes more and more bounded. It is calculated using the concept of limits in calculus and is significant in determining the convergence or divergence of infinite series. The limit cannot be negative and is related to the Pythagorean Theorem through the concept of sum of squares.
  • #1
DaTario
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Homework Statement
Let ##\sum_{i=1}^N a_{i,N} /N = 1## with ## 0 < a_{i,N} < M > 1 ## for all ##i## and ##N##.
Show that ##\lim_{N \to \infty} \sum_{i=1}^N (a_i /N)^2 = 0##.
Relevant Equations
We know that each ##a_{i,N}/N## is positive and less than one implying that their square is even smaller.
I don't know how to show that this limit is zero.
It seems that ##\sum_{i=1}^N a_{i,N} /N = 1## and the fact that ## 0 < a_{i,N} < M > 1## implies that some ##a_{i,N}## are less than one.
Another conclusion I guess is correct to draw is that ##\lim_{N \to \infty} \sum_{i=1}^N a_{i,N}^2 /N < 1##.
 
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  • #2
Try to use the fact that the ##a_{i,N}## are bounded by ##M## so that ##\left(\dfrac{a_{i,N}}{N}\right)^2## is less than some fraction times ##\dfrac{a_{i,N}}{N}##.
 
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Likes jim mcnamara
  • #3
Thank you, staddad.
 

Related to Zero Limit of Sum of Squares of Terms with Bounded Range

1. What does "Zero Limit of Sum of Squares of Terms with Bounded Range" mean?

The "Zero Limit of Sum of Squares of Terms with Bounded Range" refers to the mathematical concept of a sequence of numbers whose sum of squares approaches zero as the range of the terms in the sequence remains bounded or limited.

2. What is the significance of this concept in mathematics?

This concept is important in mathematics because it helps in understanding the behavior and properties of certain mathematical sequences, particularly those that involve squared terms. It also has applications in areas such as calculus, number theory, and statistics.

3. How is this concept related to the concept of convergence?

The "Zero Limit of Sum of Squares of Terms with Bounded Range" is a type of convergence, specifically known as convergence in the mean. This means that as the number of terms in the sequence increases, the average of the squared terms approaches zero.

4. Can you provide an example of a sequence that demonstrates this concept?

One example of a sequence that demonstrates this concept is the sequence (1/n^2) where n is a positive integer. As n increases, the terms in the sequence decrease and the sum of squares of the terms approaches zero.

5. How is this concept used in real-world applications?

This concept has various applications in real-world scenarios, such as in calculating the error in approximation methods used in numerical analysis or in understanding the behavior of electrical circuits with resistors in series or parallel. It can also be used in analyzing the convergence of certain algorithms in computer science and optimization problems in engineering.

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