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From my textbook, the definition of convergent series is given as follow.

If the sequence of partial sums {$S_n$} is convergent and $\displaystyle{\lim_{n \to \infty}} S_n$ exists, then the series $\sum a_n$ is called convergent.

So if I know that $\sum a_n$ is convergent, can I say that {$S_n$} is convergent and $\displaystyle{\lim_{n \to \infty}} S_n$ exists? As far as I know the answer is yes, but why is "if then" statement used here in the definition.

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Yes you're right.

The "if then" is used to say when the term "convergent series" applies: given any series, you can say that it is convergent if the partial sums converge. You won't call the series $\sum_n n$ convergent because the partial sums do not converge.

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The "if-then" in the sentence

If the sequence of partial sums $\{S_n\}$is convergent and $\lim_{n \to \infty} S_n$ exists, then the series $∑a_n$ is called convergent.

is an assertion about vocabulary, not a mathematical proof. It tells you when you may call a series "convergent".

The first part of that sentence is badly constructed. It says two ways that the limit of the partial sums exists. Connecting those two ways by "and" makes it sound as if there are two separate conditions when there is just one.

Much of your confusion comes from the fact that your textbook is badly written. You may encounter similar problems later on.

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  • $\begingroup$ It is actually written as limit of $\displaystyle{\lim_{n \to \infty}} S_n$ exists as a real number. $\endgroup$ – Yongyutha Kunapinan Sep 23 '18 at 13:06
  • $\begingroup$ @YongyuthaKunapinan That's somewhat more confusing. It suggests that the author considers $\infty$ a possible limit value. Sometimes we allow that informally, but this definition should not hinge on the distinction. $\endgroup$ – Ethan Bolker Sep 23 '18 at 13:33

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