1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Sequence and Series Terminology

  1. Feb 10, 2015 #1
    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.
  2. jcsd
  3. Feb 10, 2015 #2


    User Avatar
    Gold Member

    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}## ;)
  4. Feb 11, 2015 #3


    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    For that concept, see the more general (and more useful) concept of a net.
  5. Feb 11, 2015 #4


    User Avatar
    Gold Member

    Very interesting, micro.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Sequence and Series Terminology
  1. Series vs. sequence (Replies: 6)