What is the difference between a sequence and a series and how should they be used i.e. give examples of the usage of these terminologies in separate senarios.
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$\begingroup$A sequence of real numbers is a function from $\Bbb N$ to $\Bbb R$. That is a rule which assigns to each natural number $n$ a real number $a_n$.
Given a sequence of real numbers $(a_n)$ a series is defined by the sequence of its partial sums. The $n$th partial sum of the sequence $(a_n)$ is $(S_n)$ where $$S_n = \sum_{k = 1}^n a_k$$
Now, $ \sum_{n= 1}^{\infty} a_n $ is shorthand for $\lim (S_n)$. The sequence $(S_n)$ is generally called a series.
$\endgroup$ $\begingroup$A sequence, by strict definition, is a mapping from $\mathbb N$ to $\mathbb R$. Less strictly, it's just a listing of real numbers, $a_1,a_2,a_3,\dots$ For example, $$1,\frac12,\frac13,\frac14,\cdots$$ is a sequence.
A series is an infinite sum of real numbers, so for example, $$1+\frac12+\frac14+\frac18+\cdots$$ is a series.
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