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Why is the limit of an infinite series the same as limit of sequence of partial sums?

Context: I was solving a question to find the limit of an infinite series (real analysis). One method stated that it is equal to the limit of sequence of partial sums. I did not understand why.

Edit: sequence of partial sums

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closed as off-topic by Arjang, Shailesh, Hans Lundmark, Aqua, Ove Ahlman Jan 17 '18 at 13:05

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    $\begingroup$ That's by definition (otherwise the "sum of an infinite number of terms" has no meaning). $\endgroup$ – Yves Daoust Jan 17 '18 at 10:55
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    $\begingroup$ @Siddarth it might help us answer your question if you tell us a bit more: what made you think of this question in the first place? What do you think that the limit of an infinite series should be, if not the limit of its partial sums? $\endgroup$ – Omnomnomnom Jan 17 '18 at 11:07
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    $\begingroup$ It should be "sequence of partial sums" rather than "series of partial sums". $\endgroup$ – Mikhail Katz Jan 17 '18 at 11:07
  • $\begingroup$ why the downvote? It is a very good question to understand how "modern" mathematics works. $\endgroup$ – Masacroso Jan 17 '18 at 12:00
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The value of a series (any series is generally defined as an addition of infinite terms, otherwise it will be a sum) is generally defined as the value of the limit (if exists) of the sequence defined by it partial sums, that is

$$\sum_{k=m}^\infty a_k:=\lim_{n\to\infty}\sum_{k=m}^n a_k$$

Many mathematicians use the symbol "$:=$" to mean: the left hand side is defined to represent the right hand side.

Here $\sum_{k=m}^n a_k$ is a partial sum because the series is partially added up to just the $n$-th term.

If we set $x_n:=\sum_{k=m}^n a_k$ (observe the use of the symbol "$:=$") then the value of the series is equivalently defined to be the value (when it exists) of the sequence $\{x_n\}_{n\in\Bbb N}$[*].

[*] If $m> 0$ (if the series start with an index distinct of zero) then the sums of the kind $\sum_{k=m}^na_k$ are defined to have value zero when $n<m$. This is a convention called the empty sum.

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