Changing the Limits of Summation

  • Thread starter LiHJ
  • Start date
  • #1
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Homework Statement



Dear Mentors and PF helpers,

Here's my question, I see these on my textbook but couldn't really understand how they derived this short cut.
Please show me how they got to these. Thank you for your time.

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Homework Equations



These is what I understand from now.

image.jpg


The Attempt at a Solution

 

Answers and Replies

  • #2
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So we have the sum [itex] \displaystyle \sum_{r=m}^n T_r [/itex]. We may change the index of summation from r to s=r+k. Then because r starts from m, s should start from m+k and because r ends at n, s should end at n+k. This means that we changed our summation to [itex] \displaystyle \sum_{s=m+k}^{n+k}T_{s-k} [/itex]. But you should note that this sum doesn't depend on s as the first sum didn't depend on r. r and s are just dummy indices because after you do the sums, there'll remain no sign of them. So I can safely rename s to r. But because the sums are just numbers and because the changes I did to the first sum doesn't change the result, I'll have:
[itex] \displaystyle \sum_{r=m}^n T_r=\sum_{r=m+k}^{n+k}T_{r-k} [/itex].
 
  • #3
43
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Dear Shyan,

Do you mind giving me a better view with these words, can give your explanation with examples to explain these.

Thank you for your time
 
  • #4
2,793
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[itex] \displaystyle \sum_{r=m}^n T_r=T_m+T_{m+1}+T_{m+2}+\dots+T_{n-2}+T_{n-1}+T_n [/itex]

[itex] \displaystyle \sum_{r=m+k}^{n+k} T_{r-k}=T_{m+k-k}+T_{m+k+1-k}+T_{m+k+2-k}+\dots+T_{n+k-2-k}+T_{n+k-1-k}+T_{n+k-k}=\\ T_m+T_{m+1}+T_{m+2}+\dots+T_{n-2}+T_{n-1}+T_n[/itex]

Is it clear enough?
 
  • #5
43
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Thank you very much Shyan.;)
 

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