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I understand what a series is. I understand what a partial sum of a series is. But what is a sequence of partial sums? As i understand it, it's the same thing that a series.

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    $\begingroup$ Correct. The convergence divergence of a series follows that of the sequence of partial sums. This is an example of economy ot thought ubiquitous in math. We don't need to rederive the same results again and can concentrate on the new ones instead. $\endgroup$ – Jyrki Lahtonen May 11 '17 at 20:15
  • $\begingroup$ A series is two sequences: i) a sequence $a_1,a_2, \dots $ of summands, and ii) the sequence $a_1,a_1+a_2, \dots $ of partial sums. $\endgroup$ – zhw. May 11 '17 at 21:05
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Consider summming up the values $x^n$ for a fixed $x \in [-1, 1]$. So you have

1) Series

The elements are $x$, $x^2$, $x^3$, $x^4$, ...

2) Sum of the series: $$S = \sum_{n=0}^\infty x^n = \frac 1{1-x}$$

3) Partial sum up to index $n$

$$S_n = \sum_{k=0}^n x^k = \frac{x^{n+1}-1}{x-1}$$

4) Sequence of partial sums of the series

Is the sequence $S_0, S_1, S_2, \dots$. That is, is the sequence

$1, x, 1+x+x^2, \dots$ which thanks to the formula above can be written as $1, x, \frac{x^3-1}{x-1}, \dots$


Finally, note that the sum of the series is (usually, but not always) defined as the limit of the sequence of the partial sums. That is, we define the sum of the series $S$ to be

$$S = \lim_{n\to\infty} S_n = \lim_{n\to\infty} \frac{x^{n+1}-1}{x-1} = \frac 1{1-x}$$

which is indeed the expression I wrote at point 2).

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The expression: $1+2+3+4+5+\cdots$ is a series.

The sequence of partial sums of that series is: $1, 3, 6, 10, 15, \ldots$.

Thus, a sequence of partial sums is related to a series. If the sequence of partial sums converges, as a sequence, then the corresponding series is said to be convergent as well, and to equal that convergent value.

Remember: a series is a sum; a sequence is a list.

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  • $\begingroup$ A series is a series is a sequence and a sequence of partial sums is a sequence too. So why one is a sum and the other a list? 6 = 1 + 2 + 3, it's also a sum $\endgroup$ – Hugues May 11 '17 at 20:19
  • $\begingroup$ I'm not sure I understand your question. $\endgroup$ – G Tony Jacobs May 11 '17 at 20:23
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    $\begingroup$ I just mean that a series is written with plus signs, and a sequence is written with commas. $\endgroup$ – G Tony Jacobs May 11 '17 at 20:23
  • $\begingroup$ I just don't understand why is there a difference between a sequence of partial sums and a series. If we take u_(n+1) = u_n +1 (with u_0 = 0), the sequence of it's partial sum will be 1, 3, 6, 15... The series will be exactly the same thing $\endgroup$ – Hugues May 11 '17 at 20:27
  • $\begingroup$ Not really. The fourth term of the series is 4. The fourth term of the sequence is 10. It's helpful to see them as related, but semantically, we talk about them in different ways. $\endgroup$ – G Tony Jacobs May 11 '17 at 20:29
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A series is the limit of a sequence of partial sums.

From Rudin's Math Analysis, for a given sequence $\{a_n\}$, $s_n=\sum_{k=1}^n a_k$ is called the partial sum of the series. $s=\sum_{k=1}^\infty a_k$ is called the sum of the series.

The symbol $\sum_{n=1}^\infty a_n$ we call an infinite series, or just a series.

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A series $\sum u_n $ is defined by the couple $(u _n,S_n) $ where $(S_n )_{n\geq n_0}$ is the sequence of partial sums given by

$$S_n=u_{n_0}+u_{n_0+1}+...u_n .$$

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