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Even though this question deals mostly with arithmetic and geometic series, this notation is used in linear algebra and differential geomety quite a bit so I will inquire of this matter here.

What are the rules for algabreically dealing with sigma notation. When you change the value of an index from for example [itex]i=0[/itex] becomes [itex]i=1[/itex] how do you adjust the rest of the problem taking that transformation into account. Another question I have is how do you deal with injecting more into and taking quantities out of the sum, (passing through the sigma), and how that effects the rest of the sum. My final big concern is what conditions need to be met to combine / pull apart sums.

Of course any more rules or comments would be greatly appreciated.

Here is an example:

According to http://en.wikipedia.org/wiki/Evaluating_sums [Broken] for the derivation of a general rule for an geometric series they have this proof:

[tex] S = \sum_{i=0}^{n} ar^{i} [/tex]

[tex] S-rS = S(1-r) = \sum_{i=0}^{n} ar^{i} - \sum_{i=0}^{n} ar^{i+1} =

a (\sum_{i=0} ^{n} r^{i} - \sum_{i=1}^{i+1} r^{i}) = a(1-r^{n+1}) [/tex]

I get confused when [itex] \sum_{i=0}^{n} ar^{i+1} [/itex] ends up as [itex] \sum_{i=1}^{i+1} r^{i} [/itex]. How does [itex]n[/itex] become [itex] i+1 [/itex] and [itex] i=0 [/itex] becomes [itex] i=1 [/itex].

I become futher confused on how the sums dissappear into the answer, how the [itex]i[/itex] becomes an [itex]n[/itex].

What are the rules for algabreically dealing with sigma notation. When you change the value of an index from for example [itex]i=0[/itex] becomes [itex]i=1[/itex] how do you adjust the rest of the problem taking that transformation into account. Another question I have is how do you deal with injecting more into and taking quantities out of the sum, (passing through the sigma), and how that effects the rest of the sum. My final big concern is what conditions need to be met to combine / pull apart sums.

Of course any more rules or comments would be greatly appreciated.

Here is an example:

According to http://en.wikipedia.org/wiki/Evaluating_sums [Broken] for the derivation of a general rule for an geometric series they have this proof:

[tex] S = \sum_{i=0}^{n} ar^{i} [/tex]

[tex] S-rS = S(1-r) = \sum_{i=0}^{n} ar^{i} - \sum_{i=0}^{n} ar^{i+1} =

a (\sum_{i=0} ^{n} r^{i} - \sum_{i=1}^{i+1} r^{i}) = a(1-r^{n+1}) [/tex]

I get confused when [itex] \sum_{i=0}^{n} ar^{i+1} [/itex] ends up as [itex] \sum_{i=1}^{i+1} r^{i} [/itex]. How does [itex]n[/itex] become [itex] i+1 [/itex] and [itex] i=0 [/itex] becomes [itex] i=1 [/itex].

I become futher confused on how the sums dissappear into the answer, how the [itex]i[/itex] becomes an [itex]n[/itex].

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