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Hi!

While studying sequence and series, I've gotten some misunderstandings in the definitions of sequence and series.

What I know about the definitions of sequence and series is as follows below

; a sequence of field of real numbers is defined as a function mapping of the set of all positive integers into the field , and a series of real field is defined to be a sequence whose range consists of partial sums [itex]\sum[/itex] [itex]^{k}_{n=1}[/itex]a[itex]_{n}[/itex].

But, in some other books, the definitions of them are a little different from what i've written as above. I mean that in some books, they give a function from { k[itex]\in[/itex] Z : k[itex]\geq[/itex]k[itex]_{0}[/itex] for an integer k[itex]_{0}[/itex]} into the real field as a definition of a sequence.

What i'd like to ask you in this situation is whether there're no problems even if I use the different definitions of a sequence when I prove all theorems relating with the sequence ;

for example, Bolzano-weierstrass theorem, additivity of limit of two sequences, etc.

Furthermore, when we test the convergence of a series of some sepecial kinds, as you know, we usually use root, ratio tests. At first, I've learned that this tests are used when the sequence of a series runs from 1 to infinity. But, I recently found in some books that those methods are also used in determining the convergence of series running from some integer k[itex]_{0}[/itex] into infinity. That means the sequence of the series starts from the integer k[itex]_{0}[/itex].I wanna ask you if the root or ratio test can be used even in dealing with the series whose sequence runs from a integer k[itex]_{0}[/itex]. If so, I also ask you the rigorous definition of series of this kind [itex]\sum[/itex][itex]^{infinity}_{n=k_{0}}[/itex]a[itex]_{n}[/itex].

Thank you for reading my long questions.

I'll wait for your answers!!!

God bless you! Have a nice day!

While studying sequence and series, I've gotten some misunderstandings in the definitions of sequence and series.

What I know about the definitions of sequence and series is as follows below

; a sequence of field of real numbers is defined as a function mapping of the set of all positive integers into the field , and a series of real field is defined to be a sequence whose range consists of partial sums [itex]\sum[/itex] [itex]^{k}_{n=1}[/itex]a[itex]_{n}[/itex].

But, in some other books, the definitions of them are a little different from what i've written as above. I mean that in some books, they give a function from { k[itex]\in[/itex] Z : k[itex]\geq[/itex]k[itex]_{0}[/itex] for an integer k[itex]_{0}[/itex]} into the real field as a definition of a sequence.

What i'd like to ask you in this situation is whether there're no problems even if I use the different definitions of a sequence when I prove all theorems relating with the sequence ;

for example, Bolzano-weierstrass theorem, additivity of limit of two sequences, etc.

Furthermore, when we test the convergence of a series of some sepecial kinds, as you know, we usually use root, ratio tests. At first, I've learned that this tests are used when the sequence of a series runs from 1 to infinity. But, I recently found in some books that those methods are also used in determining the convergence of series running from some integer k[itex]_{0}[/itex] into infinity. That means the sequence of the series starts from the integer k[itex]_{0}[/itex].I wanna ask you if the root or ratio test can be used even in dealing with the series whose sequence runs from a integer k[itex]_{0}[/itex]. If so, I also ask you the rigorous definition of series of this kind [itex]\sum[/itex][itex]^{infinity}_{n=k_{0}}[/itex]a[itex]_{n}[/itex].

Thank you for reading my long questions.

I'll wait for your answers!!!

God bless you! Have a nice day!

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