Limit as n to infinity of sum to n -- changing upper bound to infinity? I am just wondering if, in general,
$$\lim_{n\to\infty} \sum_{i=1}^n x_i = \sum_{i=1}^\infty x_i$$
or perhaps it is specific to probability, where I am currently seeing it, such as
$$\lim_{n\to\infty} \sum_{i=1}^n P(B_i) = \sum_{i=1}^\infty P(B_i)$$
with $B_i$ all being disjoint sets, if that matters.
Is this a definition? If it is, can someone provide me a credible link that confirms it? I believe you, but my google searching is not yielding any links that say it is a definition.
If it is not a definition how could I prove it? It seems to me the standard $\epsilon$-$\delta$ definition of limit wouldn't be very helpful here, as we are not talking about the limit at a point but instead the limit is changing the number of terms summed... thats why I am leaning towards this being a definition.
Thanks
 A: It is a standard definition that
$$
\sum_{i=1}^\infty x_i = \lim_{n\to\infty} \sum_{i=1}^n x_i. \tag 1
$$
You will find this on page 59 of Principles of Mathematical Analysis, third edition, by Walter Rudin, and in many other books.
In some contexts another definition may be used, in which case the equality of those two things becomes something to be proved.  One such definition is applicable only if for every value of the index $i$, the term $x_i$ is non-negative.  It is this:
$$
\sum_{i\,\in\,\{1,2,3,\ldots\}} x_i = \sup\left\{ \sum_{i\,\in\,I} x_i : I \text{ is a finite subset of } \{1,2,3,\ldots\} \right\}. 
$$
One can extend this to a series in which some terms are non-negative and others are negative by writing
$$
\sum_{i\,:\, x_i\ge0} x_i - \sum_{i\,:\,x_i<0} (-x_i).
$$
By this definition the sum would be undefined when the sums of the positive and negative terms both diverge to infinity.  But some series in which the sums of the positive and negative terms are infinite are nonetheless well defined by $(1)$ above.  One such is the alternating harmonic series
$$
\sum_{i=1}^n \frac {(-1)^n} i.
$$
In all series in which the sums of the positive and negative terms both diverge to infinity, the value of the limit $(1)$ depends on the order in which the terms appear.
There are also alternative "summation methods" in which other definitions are used.  Perhaps the most well known of these is "Cesàro summation", named after  Ernesto Cesàro (1859–1906) according to which the sum is
$$
\lim_{n\to\infty} \frac{ x_1 + (x_1+x_2) + (x_1+x_2+x_3) +\cdots + (x_1+\cdots+x_n) } n.
$$
Every series whose sum is defined by $(1)$ above is Cesàro-summable, and the sum is the same, but some series are Cesàro-summable that are not defined by $(1)$ above; for example:
$$
1 - 1 + 1 - 1 + 1 - \cdots
$$
whose Cesàro sum is $\dfrac 1 2$.
