The Taylor-Young's formula with integral remainder

Let be a function of class and its -th derivative.

The Taylor-Young's formula

The Taylor-Young's formula tells us that any function , centered at , can be written as a Taylor series with a remainder , such as:

Here is the decomposed form:

So,

Furthermore, setting down , we do obtain a new form of this formula:

So,

Recap of the main Taylor series

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Demonstration

The Taylor-Young's formula

Starting from the main equation of the fundamental theorem of calculus:

So,

Performing an integration by parts, with a wise choice for and , we do have:

Then we do it again with:

And so on...

Thus, the function accept a Taylor series with integral remainder and this function is worth:

So,

Furthermore, setting down , we do obtain a new form of this formula:

So,

Recap of the main Taylor series

Another notation used to characterize the remainder of an Taylor series is the Landau notation .

In our specific case, we study Taylor series at the neighbourhood of , so:

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Examples

1. The sinus function

1. Taylor series of order 3 at zero

Let us use the previously demonstrated method to calculate a Taylor series of the function.

We firstly verify that can be derivated three times in a row. It is well-known that is the case.

Then, let us calculate the successive derivatives of order , and retrieve all of these images at .




Now, we apply the Taylor-Young's formula.

In our case, that would be:




2. Framing with integral remainder




We have seen above that this remainder is worth:

But, . So we now have a new expression for :




Let us now frame this remainder in the interval .

Using the property of growth of an integral, we do have:

Yet, we know thanks to that:

This leads us to a framing for :




3. Taylor series of order n at zero

Performing an Taylor series of order at for the function, we obtain:

Moreover, the remainder of this Taylor series is worth:




With the Taylor-Lagrange's inequality, we can frame this remainder:

By compared growth of the limits, the factorial function outweighs the power of x function:

In the end, with the squeeze theorem:




So as a result, a Taylor series of the function is worth: