Understanding Summation Series: Finding Patterns in Sums of Squares

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In summary, the sum of the series (1^2) + (2^2)+...(k^2) can be evaluated using the formula k(k+1)(2k+1)/6. Another method is through the use of geometric proofs, as shown in the links provided. The books "Proof Without Words" and "Discrete Calculus" are recommended for further practice and understanding.
  • #1
jobsism
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Sum of 1^2 +2^2 +...k^2

How do you evaluate the sum of the series:-

(1^2) + (2^2)+...(k^2) ?

I do know how to prove that the sum is k(k+1)(2k+1)/6 by induction, but I'm just curious: how do you figure out the pattern for this, and other similar summation series?
 
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  • #4


micromass said:
homepages.math.uic.edu/~kauffman/DCalc.pdf

Hey, thanks a lot! This is amazing. (The second link will take more time to digest, of course.)

If you are the author, you may want to fix a typo (I think) at the bottom of page 3, where [itex]12n^{\underline 3}[/itex] becomes [itex]8n(n-1)(n-2)[/itex] instead of [itex]12n(n-1)(n-2)[/itex].
 
  • #5


Thanks you, micromass and Tobias Funke! :D

I didn't even know that there was a branch of mathematics called generatingfunctionology! I didn't read much of it, but I intend to. The article on Discrete Calculus was more interesting, but it's taking some time to get used to it. Can you suggest some introductory books for it, to gain practice?

That geometric proof was simply marvellous, but such proofs can't exist for more sophisticated series, can it?
 
  • #6


jobsism said:
That geometric proof was simply marvellous, but such proofs can't exist for more sophisticated series, can it?

There are these series of books out there labeled Proof Without Words by Roger B. Nelsen which are FILLED with similar examples to this type of problem. They don't provide the proof, that's left up to you, but all of the content is viewed geometrically.

I've attached three "proofs"from the book that I thought were interesting.
 

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  • #7


Those proofs are amazing! Thanks, scurty! I'll check out that book.

The third one reminded me of the Sierpinski fractal. :D
 
  • #8


this is explained in a footnote on page 27 of courant's calculus.
 

1. What is the formula for finding the sum of 1^2 + 2^2 + ... + k^2?

The formula for finding the sum of squares from 1 to k is (k(k+1)(2k+1))/6.

2. How is the formula for the sum of squares derived?

The formula for the sum of squares can be derived using mathematical induction. It involves finding a pattern in the sum of squares from 1 to n, and then proving the pattern using mathematical induction.

3. What is the significance of the sum of squares in mathematics?

The sum of squares has many applications in mathematics, including in calculus, number theory, and statistics. It is also used to calculate the variance and standard deviation in statistics.

4. Can the formula for the sum of squares be applied to other sequences?

Yes, the formula for the sum of squares can be applied to other sequences such as cubes, fourth powers, and so on. The only difference is the exponent used in the formula.

5. Are there any real-world applications of the sum of squares formula?

Yes, the sum of squares formula has real-world applications in fields such as physics, engineering, and economics. For example, it is used to calculate the total kinetic energy of a system in physics.

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