Re: Euler's Summation Formula

Am 02.11.2006 10:39 schrieb an_idi0t@xxxxxxxxx:
Notation:

[t] is the greatest integer less than t.

Sum(f(n), y < n <=x) means sum f(n) over
integer values n greater than y and less or equal to x.

I'm trying to get a better understanding of Euler's Summation
Formula on page 54 of Tom Apostol's Analytic Number Theory.
I spent a good deal of time working through the proof and
believe I understand it line by line however I want to
understand the statement of the theorem intuitively in
a better way, if possible.

Theorem 3.1 Euler's summation formula. If f has a
continuous derivative f' on the interval [y, x],
where 0 < y < x, then

Equation (5)

sum(f(n) , y < n <= x) = int( f(t), y..x)
+ int( (t - [t]) * f'(t), y..x)
+ f(x)([x] - x) - f(y)([y] - y)


Informally, it seems to me that the sum in the LHS can
be thought of as the area of a step function from y < n <= x
where n is an integer and the value of the step function on
each interval (n, n+1) is f(n). I think the first term
of the RHS approximates the sum/step function from above.


I'm still not very familiar with the integral-expression,
although in the books of G.H. Hardy and of K.Knopp (the latter
is online available, though only in german) it is used
frequently.
I express the Euler-summation in matrix-notation, which
can easily be seen to be equivalent to Hardy's and Knopp's
sum-notation.

With a rowvector E~ = [1,1,1,1,...] and a column-vector
A = [a0,a1,a2,...an,...]~ a summation of A would simply
be written as

s = E~ * A (1)

The Euler-summation uses a replacement of E~ by a product
of a (lower triangular) binomial-matrix P

P = [1 0 0 0 ... ]
[1 1 0 0 ... ]
[1 2 1 0... ]
[1 3 3 1 ... ]
[ ... ]
and a rowvector V(1/2)~ = [1, 1/2, 1/4, 1/8,... ]

such that

1/2*V(1/2)~ * P = [1, 1, 1, 1, ... ] = E~ (2)

Now the Euler-summation uses replacement of E~ in
(1) by (2), thus establishing

E~ * A =
1/2*V(1/2)~ * P * A = s (3)

In Hardy and in Knopp the product P*A, which gives a new
column-vector, is occasionally named as B = [b0,b1,b2,...]~
such that

1/2*V(1/2)~ * B = s

and B is the Euler-transform of A.

For instance, if A = [1, -1, 1, -1, ....] then

P* A = B =

P = [1 0 0 0 ... ] * [ 1 ] = [ 1 ]
[1 1 0 0 ... ] [-1 ] [ 0 ]
[1 2 1 0... ] [ 1 ] [ 0 ]
[1 3 3 1 ... ] [-1 ] [ 0 ]
[ ... ] [...] [...]

so that B = [1,0,0,0,....]~.

Then
s = 1/2*V(1/2)~ * B (Euler-summed)
= 1/2*[1,1/2,1/4,...] * [1, 0, 0, ...]~
= 1/2

For another instance, if A = [1, -2, 4, -8, ...] then

E~ * A = s
and E~ is replaced by

1/3*V(1/3)~ * P^2 = E~
and

P^2 = [1 0 0 0 ... ] * [ 1 ] = [ 1 ]
[2 1 0 0 ... ] [-2 ] [ 0 ]
[4 4 1 0... ] [ 4 ] [ 0 ]
[8 12 6 1 ... ] [-8 ] [ 0 ]
[ ... ] [...] [...]
and

1/3*V(1/3)~ * B = 1/3*V(1/3)~ * [1,0,0,0,....]~
= 1/3

which indicates, that the Euler-Summation (E) can be
applied iteratively as (E,p) to arbitrary degree.

------------------------------------------------

If instead of 1/2*V(1/2) formally x*V(x) is written,
where V(x) contains the terms of the powerseries
V(x)~ = [1, x , x^2, x^3, ....]

then a multiplication of V(x)~ * P gives something like

V(x)~ * P = [f(x), f'(x)/1!, f"(x)/2!, ... ]
(have it not exactly at hand in the moment)

and since here derivatives occur, one can express the
Euler-summation also in terms of differentials or
integrals - but I should check this in the mentioned
books again.

Hth -

Gottfried Helms

G.H. Hardy: Divergent Series, Oxford Press
K. Knopp : Theorie und Anwendung der unendlichen Reihen (Kap XII ff)
(online at http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN378970429 )




I'm not sure what the second term of the RHS is doing,
it seems like it has something to do with the the gaps
between the step function and the approximation of the
step function from above. It seems to me that the last
two terms adjust for the non integer end points. Now
some dumb questions.

Q1 is my summary roughly what's going on? Is it a mistake
to attempt to interpret this formula in that way? Could
someone shed more light on this, and especially the second
term of the RHS.

Q2 Does this formula work if f is increasing? if f is
positive and also increasing I think the first two terms
of the RHS have to be positive so it looks to me like
the RHS will be larger than the LHS. Is that an error
in my thinking?

Q3 Is there a reason why the author used integration
variables from y to x? I think usually author's integrate
from a to b, and x to y, etc. Is integration from
y to x a hint? Is there something I'm missing about that?

Q4 If f is a positive function then - f(y)([y] - y)
is non-negative so it seems to me that |f(y)([y] - y)|
is being added to the RHS but I would expect it to be
subtracted since the sum in the LHS begins at the first
integer greater than y, and |f(y)([y] - y)| is a chunk
of the integral from some integer [y], perhaps less than
y, to y... which can't be part of the sum, can it?

Thanks.

.

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