Exploring Infinite Series: What is the Definition?

[gotjrgkr]

infinite series!

Homework Statement

Hi!
I've learned that the definition of a sequence of elements of complex numbers is as follows;
a sequence is a function whose domain is a set of all positive integers with values in a set consisting of all complex numbers. (Denote the set of all complex numbers by C from now on)
Now, let {a$_{n}$} be a sequence of elements of C.
Then, as you know, infinite series is defined as a sequence of partial sums b$_{k}$ = $\sum$$^{k}_{n=1}$a$_{n}$.If the limit of the sequence exists, then it is said that the infinite series{b$_{k}$} is convergent. In this case,
a value of the limit of the sequence is called a sum of the series and is denoted by
lim$_{k\rightarrow}$$\infty$b$_{k}$.

Now, here is my question.
I've seen a notation like this; $\sum$$_{n=p}$$^{\infty}$a$_{n}$ where p is any integer. If p is not 1, then I have no idea how to interpret this expression...
What is the exact definition of a kind of a sum above..?

Homework Equations

The Attempt at a Solution

 

[Dick]

It's $a_p+a_{p+1}+a_{p+2}+...$ It's the difference between the sum of the whole series and the (p-1)th partial sum.

 

[gotjrgkr]

It's $a_p+a_{p+1}+a_{p+2}+...$ It's the difference between the sum of the whole series and the (p-1)th partial sum.

Then what can you say about ,for example,$\sum$$^{\infty}_{n=-3}$a$_{n}$??
Do you think that it is a difference between the sum of a sequence a$_{1}$,a$_{2}$,... and (a$_{-3}$+...+a$_{0}$)?

I want to know a kind of a definition such as a form of limit expression...as I showed in the case of $\sum$$^{\infty}_{n=1}$a$_{n}$.

 

[Dick]

Then what can you say about ,for example,$\sum$$^{\infty}_{n=-3}$a$_{n}$??
Do you think that it is a difference between the sum of a sequence a$_{1}$,a$_{2}$,... and (a$_{-3}$+...+a$_{0}$)?

I want to know a kind of a definition such as a form of limit expression...as I showed in the case of $\sum$$^{\infty}_{n=1}$a$_{n}$.

You can write a series starting at any value of n, like p in the example you give. If you take a_n=1/(2^n), then the sum from n=1 to infinity is 1. The sum from n=0 to infinity is 2. The sum from n=3 to infinity is 1/4. They only differ from the sum from n=1 to infinity by a finite number of terms, as you said. If a series converges then the series starting a different limits will also converge. Just to a slightly different limit. There's no law that says the first term of a series must be a_1.