Proving Series Equations: A General Method

In summary, to prove the equation 1^{2}+2^{2}+\ldots+n^{2}=\frac{n(n+1)(2n+1)}{6} without using induction, you can use the difference equation method. However, for the similar series 1^{1}+2^{2}+\ldots+n^{n}, there is no known simple solution.
  • #1
glebovg
164
1
How to prove (not by induction)

[itex]1^{2}+2^{2}+\ldots+n^{2}=\frac{n(n+1)(2n+1)}{6}[/itex]?

What is the general approach for similar series, say, [itex]1^{1}+2^{2}+\ldots+n^{n}[/itex]?
 
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  • #2
glebovg said:
How to prove (not by induction)

[itex]1^{2}+2^{2}+\ldots+n^{2}=\frac{n(n+1)(2n+1)}{6}[/itex]?

What is the general approach for similar series, say, [itex]1^{1}+2^{2}+\ldots+n^{n}[/itex]?

For your first question, the sum is the solution to the difference equation S(n)-S(n-1)=n^2 subject to the initial condition S(1)=1. Since the difference is 2nd order polynomial, the solution is 3rd order polynomial, now you know how to proceed. For your second question, since the difference is n^n, no known simple function of n has such difference, therefore no simple solution.
 

Related to Proving Series Equations: A General Method

1. What is the purpose of proving series equations using a general method?

The purpose of proving series equations using a general method is to establish the validity of a mathematical statement or equation. It allows for a rigorous and systematic approach to verifying the truth of a series equation, ensuring that it holds true for all possible values.

2. What are the steps involved in proving a series equation using a general method?

The steps involved in proving a series equation using a general method typically include identifying the basic elements of the equation, establishing the initial conditions, setting up the inductive hypothesis, and finally, using mathematical induction to prove the equation for all possible values.

3. How does the general method differ from other methods of proving series equations?

The general method for proving series equations differs from other methods, such as direct proof or proof by contradiction, in that it uses mathematical induction to prove the equation for all possible values, rather than just a specific case or counterexample.

4. What are some common challenges in using the general method to prove series equations?

Some common challenges in using the general method to prove series equations include identifying the correct initial conditions and inductive hypothesis, as well as making sure that the steps of the proof are logically sound and follow the rules of mathematical induction.

5. Can the general method be applied to all types of series equations?

Yes, the general method can be applied to all types of series equations. It is a universal method that can be used to prove any series equation, as long as the basic elements of the equation can be identified and the initial conditions and inductive hypothesis can be established.

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