# Reversing a MacLaurin/Taylor Expansion

Discussion in 'General Math' started by Alastair, Mar 19, 2005.

1. ### AlastairGuest

Numerous functions can be expressed as an infinite series via the
MacLaurin method of expansion;

f(x)= f(0) + xf'(0) + [(x^2)/2!]f''(0) + ... + [(x^r)/r!]f^r(0) + ...
=Sum_{r = 0 to infinity}[((x^r)/r!)*f^r(0)]

{Where f^r(0) is the nth derivative of f(x) at x = 0}

For example, e^x = 1 + x + (x^2)/2! + (x^3)/3! + ...

My question is, can you take a known infinite series and reverse the
process, to produce a function which does not involve an infinite
summation?

In this case, it would be showing/proving that

1 + x + (x^2)/2! + (x^3)/3! + ... = e^x

I'm simply curious on this one, but is it possible?

Alastair, Mar 19, 2005

2. ### Jim SpriggsGuest

How would you do this other than by recognizing that an infinite series
was known by some name already? Note that there are far more infinite
series (and even far more MacLaurin expansions) than there are names in
use.

You could do it trivially by saying "I name <some infinite series> to be

Strictly speaking functions neither involve, nor fail to involve,
infinite summation. A function is just a rule for associating an
element of a set (say a real number) with the element of the same set or
another set (say another real number). The "formula" (infinite
summation or otherwise) is not the function, though in the past it was
thought that all functions must be given by formulae.
If e^x was defined to mean, let's say, lim_{n-->oo}(1 + x/n)^n, then one
might prove that that limit is 1 + x + (x^2)/2! + (x^3)/3! + ... . Is
that what you mean?

Jim Spriggs, Mar 19, 2005

3. ### AlastairGuest

My mistake. =)
My question earlier may not have been entirely clear, so I'll try
again.

This method seems to me like you will be proving it by taking e^x and
turning it into an infinite series, and showing that the two series are
the same. Imagine that you're given some infinite series, and you have
no idea how to express it without using infinite summation. For
example, if I presented you with the series such as "1 + x + (x^2)/2! +
(x^3)/3! + ... " and asked you to express it as a finite number of
terms, would there be some method (other than quoting standard
results!) of showing that the series was equivalent to some much
shorter expression? Or if the series was *not* equivalent to some
shorter expression, could this be proved?

Alastair, Mar 20, 2005