Sequence and Series Terminology and Concepts

Date: 11/27/2005 at 18:09:20
From: Ned
Subject: Sequences and series

I am attempting to work my way through Stewart's "Calculus" on my 
own, and have run into yet another source of confusion.

Stewart observes that a sequence can be defined as a function whose 
domain is the set of positive integers.  The value of the function at 
number (n), that is, f(n), is usually written a_n.  But a_n is also 
termed the "defining formula", for the sequence.  Therefore, for 
example, both the sequence {1/2, 2/3, 3/4,... n/(n + 1),...} and its 
formula n/(n + 1) are functions of (n).

However, the series (for example, a_n = 1/2 + 2/3 + 3/4+...+ n/(n + 
1) +...) is not specifically defined as a function of (n), nor is the 
sum of the series ([sum, 1 to infinity] a_n) so defined.  Yet the sum 
of the power series is said to be a function of (x) whose domain is 
the set of all (x) for which the series converges.  Is it also 
properly to be considered a function of (n) as well?
	
This can give rise to some confusion.  Consider the two series:

  [sum, 1 to infinity] x^n/sqrt n,  and

  [sum, 1 to infinity] 1/sqrt n.

The first is the sum of a power series and, therefore, a function of 
(x).  The second is the sum of a series not designated as to form, 
and not specifically described by Stewart as a function of (n).  They 
differ only in the appearance of x^n in the numerator of the first 
series.  Yet they are conceptually and importantly distinct.  Can you 
clarify the terminology for me and how these two entities are to be 
compared?  Thank you.

Date: 11/28/2005 at 02:28:17
From: Doctor Schwa
Subject: Re: Sequences and series

Hi Ned -

I'll work through your comments in order:

>Stewart observes that a sequence can be defined as a function whose 
>domain is the set of positive integers.  The value of the function at 
>number (n), that is, f(n), is usually written a_n.  But a_n is also 
>termed the "defining formula", for the sequence.  Therefore, for 
>example, both the sequence {1/2, 2/3, 3/4,... n/(n + 1),...} and its 
>formula n/(n + 1) are functions of (n).

Remember that a sequence is a list of numbers.  So which number are
you talking about?  You need to specify n in order to get a specific
number.

So, you can say the nth term is a_n, or the nth term is n/(n + 1), or
you can say the sequence is defined by the formula n/(n + 1).  Those
are all synonyms.  You just need to be careful to distinguish when
you're talking about the whole sequence (an infinite list of numbers)
versus a single term (the nth number in the list).

>However, the series (for example, a_n = 1/2 + 2/3 + 3/4+...+ n/(n + 
>1) +...) is not specifically defined as a function of (n), 

That's right!  It's not a function of n, because the series is just a
single number: the sum of the infinite collection in the list.

There's a sequence of partial sums: s_n is the sum of the first
n terms of the sequence, or

  s_n = [sum, k = 1 to infinity] a_k,

to use your notation.

Note that I introduced a new letter on the right side.  That's because
it's NOT the n of s_n.  For example, if n = 3,

  s_3 = a_1 + a_2 + a_3,

and you can see that even though n is fixed at 3, k is still a
variable, going from 1 to 3.

>nor is the sum of the series ([sum, 1 to infinity] a_n) so defined. 
>Yet the sum of the power series is said to be a function of (x) whose
>domain is the set of all (x) for which the series converges.  Is it
>also properly to be considered a function of (n) as well?

Right: suppose a_n = x/(2^n), for instance.

Then s_3 = x/2 + x/4 + x/8 is clearly a function of x.  And s_n in
general is a function of both n and x.  But the sum of all the
infinitely many terms is just 2x: not a function of n, only of x.
 
>This can give rise to some confusion.  Consider the two series:
>
>  [sum, 1 to infinity] x^n/sqrt n,  and

Notice that here the value depends only on x.  It can't depend on n:
you are summing it for all values of n.  That is, it's 

  x/1 + x^2/sqrt(2) + x^3/sqrt(3) + ...

and you'll notice there are no n's there, only x's.

>  [sum, 1 to infinity] 1/sqrt n.

This sum is just a number (well, actually it's infinity):

  1/1 + 1/sqrt(2) + 1/sqrt(3) + ...

There is no variable here at all!  The n is a "dummy variable", which
is to say it varies within the sum, but when you're done summing it's
gone.

>The first is the sum of a power series and, therefore, a function of 
>(x).  The second is the sum of a series not designated as to form, 
>and not specifically described by Stewart as a function of (n).  They 
>differ only in the appearance of x^n in the numerator of the first 
>series.  Yet they are conceptually and importantly distinct.  Can you 
>clarify the terminology for me and how these two entities are to be 
>compared?  Thank you.  

Does my explanation above help clear things up?  Please let me know if
you still have questions!

Enjoy,

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 11/28/2005 at 13:12:57
From: Ned
Subject: Sequences and series

Hello Dr. Schwa,

Thank you very much for your prompt, thorough, and extremely helpful 
response to my query.  Your observations that a series is a single 
number (which is now embarrassingly obvious to me) and the role of 
(n) as a dummy variable have clarified a murky and troublesome area 
for me.

- Ned