# Thread: Definitions of sequences and series

1. ## Definitions of sequences and series

My math books are very unclear and contradict each other regarding the defintion of a sequence and the definition of a series. I also checked wikipedia but it still isn't clear to me, so I tought I'd ask for some clarification here.

Here's what I know so far:

One of my books defines a sequence as "A function whose domain is the set of positive integers." That makes some intuitive sense, but in that case sequences can't be finite or start at a0, and that doesn't sound right. From wikipedia: "a sequence is an ordered list of objects (or events)."

Series is even more confusing. I have one book defining a series as "The sequence of partial sums {Sn} of a sequence {an}." Another one defines a series as "The sum of all the (infinite) elements of the sequence {an}." Wikipedia also mentions finite series, while my books only mentions infinite series.

Would appreciate any help with sorting this out! =)

2. My book states that a sequence is just a list of numbers written in a definite order. a1, a2, a3, a4, ... an.. where a1 is the first term and an is the nth term.

And I take it that a series is the sum of that infinite sequence if we take {An} from n=1 to infinity.

3. Part of your confusion stems from the fact that different authors use different definitions for the same terms. The standard "formal" definition of a sequence is a function from the natural numbers or positive integers - note that the range of this function is irrelevant, so you can have sequences of numbers, matrices, sets, etc. Really, no one cares if you start at 0 or 1; it's just a matter of convienence and it wouldn't be hard to turn a sequence starting at 1 into one starting at 0 while retaining the same terms. By most standard definitions, sequences are nessecarily infinite, although you can have a single term repeating infinitely many times.

With series, again, the standard way of doing it is to only consider infinite series (a finite series would just be a summation after all), which is the sequence of partial sums of a sequence of real/complex numbers (you can loosen the real/complex requirement to allow for things like matrices if you like). For example, if you have a sequence of complex numbers $a_1, a_2, ...$ then you would define $s_k = \sum_{i = 1} ^ k a_i$. Then you have another sequence, $s_1, s_2, ...$. So, series are also sequences, and they converge or diverge in the same way sequences do. The book that uses the definition of "summing up all the terms of a sequences" seems bad. What does summing infinitely many terms even mean? Without answering that question it makes no sense to define an infinite series in that way.

Both of the definitions given are the ones used by Rudin, whose texts have attained near biblical status in mathematics.

4. Keep in mind there are 2 conventions for the set of Natural numbers: one says natural numbers are {0, 1, 2, 3...} the other states the natural numbers are {1, 2, 3...} so depending on which convention the autor uses, a sequence starts at 0 or 1.

5. ## hmmmm here's how they learn me (or i didn't get it)

By the definition:
Any mapping $a: N -> R$ we'll call natural sequence. Number that with this mapping adds natural number $n$ we'll mark it with $a(n)$ or more oftenly wiht $a_n$ and call it n-th member of sequence. Number $n$ in the $a_n$ we'll call index member of sequence. For sequence $a_1,a_2,. . .,a_n,. . .$ we'll use mark $(a_n )_(n e N)$ or shorter just $(a_n )$

By the definition: (series)
Let we have sequence of real numbers $a_1,a_2,. . .,a_n,. . .$. Expression:

we'll call infinite series with common member $a_n$, or real number series.
Sums:

we'll call partial sums. For the :

we'll call n-th partial sum of the series.

That's definition witch we use sorry for bad English, hope it helped