Hi,

In our notes, our teacher explained us how to change the index of a sequence. There are some parts that I just don't understand.

He first wrote :

$\displaystyle \sum_{i=0}^{h-1}2(h-i)2^{i}=\{{i}\leftrightarrow h-i}\}=2\sum_{i=1}^{h}i2^{h-i}$

And then he "explained" (if I can call this "explain") :

$\displaystyle j=h-i$

That, I understand.

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$\displaystyle \sum_{i=0}^{h-1}2(h-i)2^{i}$

(we simply repeat the sequence)

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$\displaystyle =2\sum_{j=h}^{1}j2^{h-j}$

I understand : we move the first "2" left of Sigma, we replace "(h - i)" by "j" and "i" by "h - j"

Q1. Where does the upper limit "1" come from ?

Q2. Where does the lower limit "j=h" come from (j = h - i, not h ?!)?

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$\displaystyle =2\sum_{j=1}^{h}j2^{h-j}$

Q3. Where does the upper limit "h" come from ?

Q4. Where does the lower limit "j=1" come form ?

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$\displaystyle =2\sum_{i=1}^{h}i2^{h-i}$

Q5. Where does the lower limit "i=1" come from ? It's upposed to be "i = j - 1", so if we previously had "j = 1", we should have "i = 1 - 1 = 0".

Q6. Why has the "j" between Sigma and "2" become "i" ?

Q7. Why the exponent "h-j" has become "h - i" ?

Am I simply dumb or our teacher forgot to write a few things ?

Thanks for your help