March 27, 2018 Tuesday
Bedtime Story
Note G of Ada Lovelace - 5
We are continuing tonight with Note G of
Ada Lovelace.
“As however our object is not simplicity or
facilitation of computation, but the illustration of the powers of the engine,
we prefer selecting a formula below, marked (8).
This is derived in the following manner: –
If in the equation
Numbered as (4.)
(in which B1, B3… etc are the Numbers of
Bernoulli), we expand the denominator of the first side in powers of x, and
then divide both numerator and denominator by x, we shall derive
Numbered as (5.)
If this latter multiplication be actually
performed, we shall have a series of the general form
1 + D1x + D2x2
+ D3x3 + … (6.)
In which we see, first, that all the
coefficients of the powers of x are severally equal to zero; and secondly, that
the general form for D2n, the coefficient of the 2n+1th term (that is of x2n
and even powers of x), is the following:-
Numbered as (7.)
Multiplying every term by (2.3…2n) we have
Numbered as (8.)
Which it may be convenient to write under
the general form:-
0 = A0 + A1B1
+ A3B3 + A5B5 +…+ B2n-1 (9.)
A1, A3 etc being those
functions of n which respectively belong to B1, B3 etc.
We might have derived a form nearly similar
to (8.), from D2n-1 the coefficient of any of power of x in (6.),
but the general form is a little different for the coefficients of the odd
power, and not quite so convenient.
On examining (7.) and (8.), we perceive
that, when these formulae are isolated from (6.), whence they are derived, and
considered in themselves separately and independently, n may be any whole
number whatever; although when (7.) occurs as one of the D’s in (6.), it is
obvious that n is then not arbitrary, but is always a certain function of the distance of that D from the beginning.”
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic engineer
and a former professor at BARC, does and teaches real mathematics and physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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