Leibniz formula using mathematical induction

In summary, the Leibniz's rule states that if you have a function that takes in a finite set of input values and outputs a finite set of output values, then you can sum up the input values without having to worry about the order in which they were given. If you're familiar with the substitution rule for integrals, this is basically similar, but without the need for limits.
  • #1
Jkohn
31
0

Homework Statement


Here is this problem:
IMG_20121206_202225.jpg


IMG_20121206_202342.jpg


I have the solution http://www.proofwiki.org/wiki/Leibniz%27s_Rule/One_Variable

This is where I get stuck..
Where it says: 'For the first summation, we separate the case k=n and then shift the indices up by 1.'

Why does this lead to the conclusion??
thanks
 
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  • #2
What they do is basically

[tex]\sum_{k=0}^n a_k = a_n+ \sum_{k=0}^{n-1} a_k = a_n + \sum_{k=1}^n a_{k-1}[/tex]

Do you understand these steps?
 
  • #3
micromass said:
What they do is basically

[tex]\sum_{k=0}^n a_k = a_n+ \sum_{k=0}^{n-1} a_k = a_n + \sum_{k=1}^n a_{k-1}[/tex]

Do you understand these steps?

Interesting, I didnt know of this rule..whats it called??
 
  • #4
Jkohn said:
Interesting, I didnt know of this rule..whats it called??

It's not really called anything.

If you're familiar to the substitution rule for integrals, then this is pretty similar.
In general: if [itex]f:A\rightarrow B[/itex] is a bijection between the finite sets A and B, then

[tex]\sum_{k\in B} a_k = \sum_{k\in A} a_{f(k)}[/tex]

But most important: do you see why the equalities hold??
 
  • #5
No I dont.
 
  • #6
Jkohn said:
No I dont.

Perhaps try to write out the summations?? So in

[tex]\sum_{k=0}^n a_k = a_n + \sum_{k=1}^n a_{k-1}[/tex]

replace all summation signs by the actual sums.
 
  • #7
micromass said:
What they do is basically

[tex]\sum_{k=0}^n a_k = a_n+ \sum_{k=0}^{n-1} a_k = a_n + \sum_{k=1}^n a_{k-1}[/tex]

Do you understand these steps?

Jkohn said:
No I dont.

It's really not very complicated. From the first sum to the next step, all that has happened is that they have separated out an from the summation. The first summation has n + 1 terms, a0, a1, ..., an. The expression in the middle has an and a summation with n terms, a0, a1, ..., an-1. In all, it's the same (n + 1) terms as in the first summation.

Going from the expression in the middle to the last one, all they are doing is fiddling with (i.e., adding 1 to) the index, where the index ranges between 1 and n instead of between 0 and n - 1 as before. To adjust for this, the subscript on a is adjusted correspondingly.

In all three expressions, they are adding n + 1 terms, a0, a1, ..., an.
 
  • #8
ohhhhh makes sense..thanks!
 

1. What is the Leibniz formula using mathematical induction?

The Leibniz formula using mathematical induction is a method for proving the validity of the formula for the sum of an infinite alternating series. It states that if a series is alternating and its terms approach zero, then the sum of the series can be found by taking the limit of the partial sums.

2. How does mathematical induction apply to the Leibniz formula?

Mathematical induction is used to prove that the Leibniz formula is true for all values of n, where n is the number of terms in the series. This is done by showing that the formula holds for n = 1, and then assuming that it holds for n = k and proving that it also holds for n = k+1.

3. What are the steps for using mathematical induction to prove the Leibniz formula?

The steps for using mathematical induction to prove the Leibniz formula are as follows:

  1. Prove that the formula holds for n = 1.
  2. Assume that the formula holds for n = k, where k is some positive integer.
  3. Using the assumption, prove that the formula also holds for n = k+1.
  4. Conclude that the formula holds for all positive integers n by the principle of mathematical induction.

4. What is the significance of the Leibniz formula using mathematical induction?

The Leibniz formula using mathematical induction is significant because it provides a rigorous method for proving the validity of the formula for the sum of an infinite alternating series. This formula is also widely used in mathematics and physics to approximate values of infinite series.

5. Are there any limitations to using mathematical induction to prove the Leibniz formula?

Yes, there are limitations to using mathematical induction to prove the Leibniz formula. This method can only be used to prove the formula for positive integers, and it may not work for more complex series. Additionally, it requires a strong understanding of algebra and mathematical reasoning to successfully apply it.

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