| Version 7 |
Version 6 |
| If a series $\sum_{n = 1}^\infty a_n$ of real or complex numbers is convergent and the limit of its partial sums is $S$, then $S$ is the {\em sum of the series}. \,This circumstance may be denoted by |
If a series $\sum_{n = 1}^\infty a_n$ of real or complex numbers is convergent and the limit of its partial sums is $S$, then $S$ is the {\em sum of the series}. \,This circumstance may be denoted by |
| $$\sum_{n = 1}^\infty a_n = S$$ |
$$\sum_{n = 1}^\infty a_n = S$$ |
| or equivalently |
or equivalently |
| $$a_1+a_2+a_3+... = S.$$ |
$$a_1+a_2+a_3+... = S.$$ |
| Nevertheless, one should not think that this means an addition of infinitely many numbers --- it's only a question of the limit |
Nevertheless, one should not think that this means an addition of infinitely many numbers --- it's only a question of the limit |
| $$\lim_{n\to\infty}\underbrace{(a_1+a_2+...+a_n)}_{\textrm{partial sum}}.$$ |
$$\lim_{n\to\infty}\underbrace{(a_1+a_2+...+a_n)}_{\textrm{partial sum}}.$$ |
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| The sum of the series is equal to the sum of a partial sum and the corresponding remainder term. |
The sum of the series is equal to the sum of a partial sum and the corresponding remainder term. |
