Proving an Identity Using Series Manipulation

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In summary, the conversation discusses a problem with proving an identity involving J_p(x) and the differential of x^{-p}J_p(x). The solution involves changing the index of summation in one of the expressions to make them match.
  • #1
nille40
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Hi everyone!
I'm trying to prove an identity, and it's driving me insane.

Let [tex]J_p(x) = \sum_{n=0}^{\infty} \left(-1\right)^n\frac{x^{2n+p}}{2^{2n+p}n!(n+p)!}[/tex]

Show that
[tex]\frac{d}{dx}(x^{-p}J_p(x)) = x^{-p}J_{p+1}(x)[/tex]

I get the left part to
[tex]\sum_{n=0}^\infty(-1)^n\frac{x^{2n-1}}{2^{2n+p-1}(n-1)!(n+p)!}[/tex]

And the right part to
[tex]\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{2^{2n+p+1}n!(n+p+1)!}[/tex]

This is incorrect, since they are not equal. Please, please help me! I can post my calculations if that would help.

Thanks in advance,
Nille
 
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  • #2
You might notice that the lower index on
[tex]\sum_{n=0}^\infty(-1)^n\frac{x^{2n-1}}{2^{2n+p-1}(n-1)!(n+p)!}[/tex]
is incorrect. The first term will be n= 1, not n=0.

Try changing the index from n to j= n-1 so that the sum will be
from j=0 to infinity.

If j= n-1 then n= j+1 so, for example, the power of x, 2n-1, becomes 2(j+1)-1= 2j+1. Make that change for every n in the formula.

Of course, since n (or j) is a "dummy" variable, you could then just replace "j" with "n" or change the n in the second sum you have with j in order to compare them.
 
  • #3
That's it!
I had a hard time realizing that that you could just change the index. But I realize that "infinity - 1" is, obviously, infity. Hince the delay...

Thanks for helping me!
Nille
 

What is an identity in mathematics?

An identity in mathematics is an equation that is true for all values of the variables involved. It is a statement of equality that holds true regardless of the values assigned to the variables.

Why is it important to prove an identity?

Proving an identity is important because it allows us to verify the validity of mathematical statements and equations. It also helps us understand the relationships between different mathematical expressions and provides a foundation for more complex mathematical concepts.

What are the common methods for proving an identity?

The most common methods for proving an identity include direct proof, indirect proof, proof by contradiction, and proof by induction. Direct proof involves showing that both sides of the equation are equivalent, while indirect proof involves assuming the opposite of what is to be proven and then showing that it leads to a contradiction. Proof by contradiction involves assuming that the identity is false and then showing that it leads to a contradiction, and proof by induction involves proving the identity for a base case and then showing that it holds true for all other cases.

What are some tips for successfully proving an identity?

Some tips for successfully proving an identity include carefully examining the given equation and identifying any patterns or relationships, using known identities and properties to simplify the equation, and approaching the proof with a clear and organized strategy. It is also important to double-check all steps and make sure they are logically sound.

What are some common challenges when trying to prove an identity?

Some common challenges when trying to prove an identity include getting stuck at a certain step and not being able to move forward, making careless mistakes, and not having a clear understanding of the properties and identities being used. It is also common to encounter unfamiliar or complex identities that require a deeper understanding of mathematical concepts.

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