Using Telescoping Property for Summing ∑(2k-1)

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In summary, the telescoping property allows us to find the sum of a finite sequence of numbers by using the principle of addition and the telescoping sum. This is a reliable method of reasoning about finite sums.
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unintuit
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1+3+5+...+(2n-1)=∑(2k-1)

but (2k-1)=k2-(k-1)2
summing we use the telescoping property and deduce that ∑(2k-1)=n2-02=n2

This seems accurate to me. Now my question is this a proper use of the telescoping property. In the least it reveals the proper answer, which can then be proved by induction.
 
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  • #2
You example illustrates the use of a telescoping sum to reach a correct conclusion about finite sums. This is a reliable method of reasoning about finite sums. Is that what you mean by the words "accurate" and "proper" ?

The problem of finding simple expressions for finite sums can be regarded as the problem of "anti-differencing", which has an analogy to antidifferentiation in calculus. You can find material online about doing anti-differencing and there are rules for it that are analogous to those used for antidifferentiation. For example, there is Summation By Parts ( http://en.wikipedia.org/wiki/Summation_by_parts ) which is analgous to Integration By Parts.
 
  • #3
Yes to your first question. I was also interested in how one would determine the limit of the expression Σ[k2-(k-1)2]=Σ(2k-1). How would one know if it were to approach 0 vs. 1.
I assumed that it went to zero because
i. (k+1)2=k2+2k+1=(k+1)2-k2=2k+1 then summing over n we have [(n+1)2-(1+n)]/2=∑k. subtracting 1 from the terms in i. gave [k2-(k-1)2]→(n+1-1)2-(1-1)2=n2. I did not know how to determine the limit which is why I was curious if this is a good way to figure out the sum vs. some unknown way. If there is some other way, what would it be?
 
  • #4
unintuit said:
Yes to your first question. I was also interested in how one would determine the limit of the expression Σ[k2-(k-1)2]=Σ(2k-1).

What limit are you talking about? The limit of the expression as k approaches infinity? - or the limit of the expression as k approaches zero? - or as k approaches some other number?
 
  • #5
Nevermind, I figured out my mistake. I was thinking about it in the wrong way. It was starting with k=1 and ki∈ℕ with k1<k2.
Thank you for your help.
 

What is the telescoping property for summing?

The telescoping property is a mathematical technique used to simplify the sum of a series by canceling out terms. It involves grouping terms in a way that allows for cancellation and reduces the number of terms in the sum.

How does the telescoping property work?

The telescoping property works by grouping terms in a way that allows for cancellation. This is typically done by finding a pattern in the terms and using it to simplify the sum. In the case of ∑(2k-1), the pattern is that every other term is cancelled out, leaving only the first and last terms in the sum.

What is the formula for using the telescoping property to sum ∑(2k-1)?

The formula for using the telescoping property to sum ∑(2k-1) is: ∑(2k-1) = 1 + (2n-1), where n is the number of terms in the series.

What are the advantages of using the telescoping property for summing?

The main advantage of using the telescoping property for summing is that it simplifies the sum and reduces the number of terms, making it easier to calculate. It also allows for patterns and relationships between terms to be identified, which can be useful in other mathematical calculations.

What are some common applications of using the telescoping property for summing?

The telescoping property can be used in various mathematical and scientific fields, such as physics, engineering, and finance. It is often used to simplify complicated sums and series, making them easier to work with and analyze. It can also be applied in real-life situations, such as in calculating the total cost of a project or investment.

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