Sequence and Series Terminology

[Mr Davis 97]

First, I would like to clear up notation and the definition for sequences. What exactly is a sequence? I read somewhere that it is defined as a function $f: \mathbb{N} \to \mathbb{R}$. But if this is the case, why do we only define functions based on the range of the function, e.g., $\left \{ 1, 4, 9, 16... \right \}$ (which we regard as "the sequence")? We define sequences with the notation $\left \{ a_{n} \right \}_{0}^{\infty}$ too, but what does this mean in terms of the functions concept? How does this specify a function? In addition, what is the nth term's relation to the function concept, or in other words, what is the analogue of the nth term for sequences in functions? Finally, what exactly to the terms "partial sums, series, finite series, infinite series," mean? It seems as though they are mostly for the same concept.

 

[BiGyElLoWhAt]

I like the function definition. The reason we take integers is, honestly, for simplicity. You plug in 1, and you get your first term, you plug in 2, and you get your second term, etc. If you like, I suppose you could do 1/2 integers, and 1/2 would be your first term, 1 would be your second term, etc. You could also do multiples of pi/6. You plug in pi/6 and you get your first term, pi/3 and get your second term, etc. The function would have to be defined accordingly though. IMHO, just stick with the integers, because that is consistent with other definitions. If you were going to define your series to be $\{ a_n\}_0^\infty\ \text{where}\ a_n := f(n)$ and you took n to be something other than integers, what would your pi'th term be? You see how it gets conceptually sketchy there?
Partial sums are when you do a sum over a well defined range, i.e. a finite series. Ex. $\sum\limits_{k=1}^{10} k$ This is a finite series/partial sum. The equivalent infinite series would be $\sum\limits_{k=1}^{\infty} k = -\frac{1}{12}$ ;)

 

[micromass]

If you like, I suppose you could do 1/2 integers, and 1/2 would be your first term, 1 would be your second term, etc. You could also do multiples of pi/6. You plug in pi/6 and you get your first term, pi/3 and get your second term, etc. The function would have to be defined accordingly though.

For that concept, see the more general (and more useful) concept of a net.

 

[BiGyElLoWhAt]

For that concept, see the more general (and more useful) concept of a net.

Very interesting, micro.