What is a formal definition of series? Is there a formal definition for series?
For example, cardinal sum has a formal definition such that $\sum a_i$ = $\bigcup a_i$.
Is there any clear definition for series of real or complex number?
The definition on my book is the sum of $a_0 + ... + a_n$. This seems very intuitive to me so i don't like it..
I tried to define series such that $\gamma(0) = a_0$ and $\gamma(n+1) = f_n(\gamma(n))$ where $f_n(x) = x+ a_n$
Since $f_n ≠ f_{n+1}$, i can't apply finite recurssion theorem. However it seems obvious that $\gamma$ is a function and unique. How do i prove the existence and uniqueness of $\gamma$?

I have proved that "If A is a set, 'c' a fixed point in A and $f_n : A →A$ a function for every $n\in \mathbb{N}$, then there exists a unique function $\gamma : \mathbb{N} →A$ such that $\gamma(0) = c$ and $\gamma(n+1) = f_n(\gamma(n))$. This is a bit generalized form of original finite recursion theorem.
Let $\alpha$ be a sequence. Let $f_n(x)=x+\alpha(n+1) : F→F$. (F denotes an arbitrary field here) Then by above theorem, we can construct $\gamma$. It can be easily checked that $\gamma(n)$ is a summation of $a_0,...,a_n$.
By the way, i've never said series is a 'finite summation'. Of course, series is a sequence..
 A: $$\sum_{i=1}^2a_i=a_1+a_2$$
$$\sum_{i=1}^{n+1}a_i=\sum_{i=1}^na_i+a_{n+1}$$
$$\sum_{i=1}^{\infty}a_i=\lim_{n\to\infty}\sum_{i=1}^na_i$$ if the limit exists. 
A: Take any sequence $(a_0,a_0,\ldots)$ of real numbers. Define a function $r$ on finite sequences of real numbers such that if $s$ has length $n-1>0$, $r(s)$ is the last value of $s$ plus $a_n$. If the length of $s$ is $0$ (which we allow), then $r(s)=a_0$. The (countable) recursion theorem gives you a unique sequence $(b_0,b_1\ldots)$ such that $b_{n+1}=b_n+a_{n+1}$ and that defines your series. Note that the recursion is allowed to depend on all finite sequence of values.
There are two things you might think are not rigorous here: The "length" and the "last value". A finite sequence is a function with domain being a finite ordinal. This ordinal is the length, which is therefore well-defined. the elements of an ordinal are again ordinals and the value at the largest such ordinal in the domain is the "last value".
A: To answer the question in the title, a series is a pair of sequences $(x_n)_{n\geqslant0}$ and $(S_n)_{n\geqslant0}$ such that $S_n=\sum\limits_{k=0}^nx_k$ for every $n\geqslant0$ (with the obvious modifications for sequences and series indexed by $n\geqslant i$ with $i\ne0$).
