What is the definition of series?
$$\sum_{i=0}^\infty i = 1+2+3+4\,+..$$
Would you call the left hand side "series"? Because the right hand side is the sum of the series and i don't understand why people seem to say series = sum of the series. Seems like a simplification.
I believe I was taught that infinite series is a sequence of growing partial sums of another sequence. For example:
$$series\;example = (1, 1+2, 1+2+3, 1+2+3+4,…) = (1,3,6,10,15,21,…)$$
Are series some sort of sequences or are they inifnite sums?
Best Answer
The series in the form $\sum_{i=1}^{\infty}i$ admits two different interpretations.
It can be seen as sequence of partial sums \begin{align*} \sum_{i=1}^{\infty}i:= \left(1,3,6,\ldots\right) =\left(\sum_{i=1}^n i\right)_{n\geq 1} \end{align*}
and it can be seen as limit of the sequence of partial sums \begin{align*} \sum_{i=1}^{\infty}i:=\lim_{n\to\infty}\sum_{i=1}^n i \end{align*}
Note: You can find this line of argumentation e.g. in Theory and Application of Infinite Series by K. Knopp.