I noticed that a Taylor series, when you add up all values from 0 to infinity, is similar to the Riemann sum. I also know that the Riemann sum has change of x as a critical component and, as far as I can tell, the Taylor series formula doesn’t.
Is there a way to manipulate the Taylor series formula into an integral?
I assume that you want to express a Taylor series as a Riemann integral? Any infinite sum can be expressed as a Lebesgue integral with respect to counting measure. And of course you can also express a Taylor series as an integral by differentiating each term of the Taylor series and then integrating the resulting infinite sum, but I assume that is also not what you want. As far as I know there is no general way to express a Taylor series as an integral in a closed form without relying on another infinite sum.
You have to explain why the infinite \sum a_nx^n resembles \lim\sum f(x#i)(x_i-x{i-1} where x#i is in the interval [x{i-1},x_i]. I just don’t see it.
We used to have handbooks that gave particular equalities between integerals and series, some of which were Taylor series. Except for special cases, we never ‘manipulated’ a Taylor series into an integral, (but we wouldn’t have done so even if it had been possible).
Every series can be converted directly to an integral, because if nothing else, you can make a stairstep function with a discontinuity at every integer, and with the height of each step being the value of that term of the series. The integral of that function will then be the sum of the series.
This isn’t particularly useful, of course, because any method to solve that integral will start by converting it back to a series. Though you can often use that stairstep function to construct a pair of smooth functions, one less than it and one greater, and then integrate those functions, in order to prove convergence of the series, as well as bounds on what it converges to.
Examples might be useful; I think he or she is talking about formulae like
∞
⌠ cos(t)
-⎮ ────── ⋅ dt
⌡ t
x
=
∞
_____
╲
╲ k
╲ ⎛ 2⎞
╱ ⎝-x ⎠
γ + ln(x) + ╱ ───────
╱ 2k(2k)!
‾‾‾‾‾
k = 1
What you are suggesting is of course useful and can be expressed in various forms like
p
=====
\
> f(i)
/
=====
i = a + 1
=
b
/
|
| f(x) dx
|
/
a
+
B
__ p k (k - 1) (k - 1)
\ --(f (b) - f (a))
/__ k = 1 k!
+
...
where the stairstep function (Bernoulli polynomial evaluated at the fractional part of x) occurs in the last term I did not write out
It is useful when you wish to do a multiplication or convolution on the integral.