Definition of sequence and series

[gotjrgkr]

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 $\sum$ $^{k}_{n=1}$a$_{n}$.
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$\in$ Z : k$\geq$k$_{0}$ for an integer k$_{0}$} 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$_{0}$ into infinity. That means the sequence of the series starts from the integer k$_{0}$.I want to ask you if the root or ratio test can be used even in dealing with the series whose sequence runs from a integer k$_{0}$. If so, I also ask you the rigorous definition of series of this kind $\sum$$^{infinity}_{n=k_{0}}$a$_{n}$.

Thank you for reading my long questions.
I'll wait for your answers!
God bless you! Have a nice day!

 

[CompuChip]

Whether you start your sequence at 0, 1 or any other positive integer k₀ does not matter. It is an easy exercise to prove that they are equivalent (basically finding a bijection is enough - and the bijection is just shifting the index by k₀).

You can define an "infinite" series as the limit of the partial sums:
$$\sum_{n = 0}^\infty a_n = \lim_{N \to \infty} \sum_{n = 0}^N a_n$$

 

[gotjrgkr]

Do you mean that I can reach to the same theorem (but the definition of sequence in the theorem is a little different) concerned with a sequence even though I use different definitions of a sequence ??

In the above equation, does the symbol N run from 0? or 1?

 

[CompuChip]

Yes, because I claim that the definition is equivalent for all practical purposes as they are related by a simple shift.

I don't really understand the second question... N is the limit variable.

 

[gotjrgkr]

Yes, because I claim that the definition is equivalent for all practical purposes as they are related by a simple shift.

I don't really understand the second question... N is the limit variable.

What I know about the definition of limit of a given sequence <b$_{N}$>( it starts from 1;b$_{1}$,b$_{2}$, ...) is as follows; the sequence <b$_{N}$> converges to b$_{0}$ if for any $\epsilon$>0, $\exists$M>0 such that N>M implies $\left|$b$_{N}$-b$_{0}$$\right|$<$\epsilon$.
In the above equation you show me, it's a limit of a sequence <b$_{N}$>=$\sum$$^{N}_{n=0}$a$_{n}$. If the sequence <b$_{N}$> starts from 1, then I can apply the definition of limit of a sequence to this sequence. But if the sequence <b$_{N}$> starts from 0, then it's confusing for me to apply the definition of limit of sequence as I mentioned above. But You told me that it doesn't matter whether I use different definitions of a sequence. So I wonder if in the case of the sequence <b$_{N}$> there's another definition of limit of sequence, too. ;;Because of this I'm confused .