A new way to represent functions as series

Abstract In this paper we will show a new way to represent functions as infinite series, finding some conditions under which a function is expandable with this method, and showing how it allows us to find the values of many interesting series. At the end, we will prove one of the main results of the paper, a Representation Theorem.


Introduction
It has always been an interesting problem to nd the sums of in nite series. There are many examples of series whose values are known: Taylor Series, Fourier Series, ... (see, for example, the list of references at the end of this manuscript).
Here our aim is to show a new method to expand functions as series. In the rst sections, we present how a particular iteration of Lagrange's mean value theorem leads us to a new series expansion. Then, we nd some conditions of expandability for some common functions. In the last section, we develop a more general theory, which gives us the values of other series.

A comparison with other well known series
Before introducing this new kind of expansion, we will analyse the conditions of expandability of Taylor series and Fourier series, comparing them with the new results presented here.

. Taylor series
This kind of series allows us to represent a function f as an "in nite polynomial". More precisely, we call power series a series of this type: It is apparent that L ≥ ; the convergence radius of the power series is de ned as R := L , using the convention / = ∞ and /∞ = if either L = , or L = ∞. Then the power series converges ∀x ∈ (x − R, x + R), whereas at the extrema of the interval, the convergence depends on the particular series considered.
We say that a function f is analytic in I = (a, b) if ∀x ∈ I it can be written as a power series with center x and a proper convergence radius R > . Such a power series is usually known as Taylor series. If a function is analytic, then it turns out that the coe cients a k are directly linked to the derivative of order k of f evaluated at x , namely, moreover, the series converges pointwise to f (x) when x is in a suitable neighborhood of x (a Taylor series with x = is also called a McLaurin series). It is also clear that, if f is analytic in I, then f ∈ C ∞ (I). .

Fourier series
Let f : R → R be a periodic function with period π; in such a case, under mild conditions on f , Fourier series allows us to associate an "in nite trigonometrical polynomial" to f . We can write where at this stage ∼ simply means that the series is associated to the function f , since Although we assumed that the period is π, the theory can be generalized to any period T > ; see, for example, [9]. When dealing with Fourier series, we can consider di erent notions of convergence. Here we will consider only two of them. The rst one is the norm convergence: let L = L (−π, π) denote the Hilbert Space of squareintegrable functions over the interval (−π, π). If f ∈ L , then when n → +∞, where Sn f = a + n k= (a k cos(kx) + b k sin(kx)). On the other hand, we can consider the pointwise convergence of Fourier Series: suppose that f is a bounded function in [ , π] and that this interval can be decomposed as a nite number of subintervals, such that in each of them f is continuous and di erentiable. Suppose also that the limits of f and f at the extrema of these subintervals are nite. Then, if we let f (x ± ) denote the right/left limit of the function at x , we have .

A new kind of expansion
The new kind of expansion we develop here requires two fundamental conditions: f ∈ C ∞ ([x − , x + ]) and the one in Proposition 4.1. We will analyse some special cases of this Proposition, which will lead us to many interesting results. Using this method, the function f can be represented with a series of products depending on the inverses of the derivatives f (i) . Eventually, in Section 8, we develop a more general method, that relaxes the previous assumptions on f , and does not require the di erentiability of f ; as a matter of fact, in that case we can nd a general representation for any real number x.
We can conclude that -Taylor series require strong conditions on f , but ensure good convergence properties; -Fourier series require weak conditions on f , but convergence needs to be understood in a proper sense; -The rst kind of series developed here requires strong conditions on f , namely f ∈ C ∞ , whereas the second one requires weaker conditions.

A particular case
We will start with an example related to a general smooth function in the interval [ , ]. First of all, recall that a function f is smooth when it has derivatives of all orders. If a function is smooth in an interval [a, b] ). Consider the interval (2,3); since f is continuous (also at the extrema 2 and 3) and di erentiable in it, for Lagrange's mean value theorem we have: for a point c ∈ ( , ). For the same reason, For what we said, we now know that f (c ) + f (c ) = f ( ) − f ( ). Now we want to nd the di erence between f (c ) and f (c ), so that, with a system, we can nd their values. To do this, we use again Lagrange's theorem Putting in a system this equation with the one with the sum of f at c , and solving, we obtain and Now, always for the mean value theorem, we have that (∀h ≥ ) Knowing that f (c ) − f (c ) = (c − c )f ( ) (c ) (obtained from the above formula with h = ), we can put this value into (3.6) and get

Theorem 3.1. Let c and c de ned as above. Then
If h = , we obtain (3.6).
Proof. The proof is by induction. We know that for h = the formula holds true. Now, suppose that it holds true for a number h. We have to prove that it also holds for h + . Then, by induction, we can conclude that the formula holds true ∀h ≥ . So, suppose that Consider now this formula for h + : This can be written as and it implies that Hence, we have that Since the rst expression equals f ( ), also the second is equal to f ( ), and therefore, we proved the theorem.
Instead of considering the intervals ( , ) and ( , ), we can take (x − , x ) and (x , x + ) for any x ∈ R.
In the same way as in Theorem 3.1, we then get the following result.
Proof. Analogous to the one of the previous Theorem.
Notice that to nd the values of all the c j , we have to solve

Series associated to a function
To show the next results, we will use the following notation: De nition 4.1. Let c j be the points de ned before. For j ≥ we will let l(x , j) := c j − c (l depends on j and x in general), Now our aim is to take the limit for h → +∞ of the expansion in Theorem 3.2. More precisely, we want to understand under which conditions, we can say that In the following, when we say that a function is expandable, we mean that can be written as a series above.
Our aim is to nd out when a function can be written as in (4.1), and we also want to determine the expansions of some of the principal functions of Analysis.
We will start with the following.

Proposition 4.1. Let f be a function with the properties in Theorem 3.2. Then f is expandable if and only if
Proof. From Theorem 3.2, we know that (x , i)).
in order to obtain (4.1), we must have Equation (4.1) can be expressed also in this way.
This can be represented in another interesting way From the formulae that give c j , we know that Iterating this, we obtain the above theorem.

Expandability of a function
We now want to nd some theorems that can be easily used to determine when a given function f can be expanded in series. To prove the rst one, we need the following: Generalizing this, we can say that c i+j < c i , ∀i ≥ , ∀j ≥ . So, if l(x , i ) ≤ , this inequality holds true ∀i ≥ i .
Since |f (i) (x)| → ∀x ∈ (x − , x + ), and c h ∈ (x − , x + ), we obtain |f (h− ) (c h )| → ; we have to show that n(x , h − ) ∞, so that the product goes to . Since l(x , i ) ≤ , by Lemma 5.1 l(x , i) ≤ ∀i ≥ i . Hence, We would like to have also some conditions under which n(x , i) → , since in general we cannot write a closed formula for this quantity. In order to do this, we need a theorem about in nite products. Therefore, +∞ n= log(l(x , n)) = −∞. Applying Theorem 5.4, we conclude. We end with this important theorem.
because under the said hypothesis n(x , h − ) → by Theorem 5.5.

Expansions of some important functions
In this section we will show the expansions of some functions. Example 6.1. Let f (x) = e ax , a ∈ R, a ∈ (− , ), a ≠ . |f (i) (x)| = |a| i e ax → . We want to nd some of the c h . To do this, we have to solve .
As noticed before, l(x , ) = and |f (i) (x)| → ∀a ∈ (− , ), a ≠ , so by Theorem 5.1, f is expandable for these values of a. To write its series, notice that Now, notice that when a = ± , |f (i) (x)| = e ±x ≤ M ∈ R+ since it is independent of i. We wonder, whether the function is expandable also for these values, or not. We know that the derivatives are limited, so we just have to verify that ∃j : l(x , j) < e in both cases. Take, for instance, j = ; we have , if a = − .
Since both these values are < e , we can expand f . We get a new interesting representation of e, i.e. , ω ∈ ( , ). In this interval, we have that |f (i) (x)| ≤ |ω| i → when i → +∞.
To know when f is expandable and to write its series, we evaluate some points c j ; notice that we can choose any x ; here, for the sake of simplicity, we will just consider the particular case x = . We have where p , p ∈ Z since cos y = t for < t ≤ yields y = pπ ± arccos t for some p ∈ Z, and c ∈ ( , ), c ∈ ( , ) because of Lagrange's Theorem. So we want to nd p , p ∈ Z such that c ∈ ( , ) and c ∈ ( , ). We claim that they are both equal to , and that we have to choose the sign +. To prove this, we have to verify that < ω arccos( sin ω ω ) < , First of all, notice that the function h(ω) := sin ω − ω cos ω is strictly increasing in ( , ), and h( ) = ; therefore, sin ω − ω cos ω > in ( , ) and sin ω ω > cos ω. Since arccos is a decreasing function in its domain of de nition, we have arccos( sin ω ω ) < ω. Noting also that arccos y = only when y = , and that sin ω ω ≠ when ω ∈ ( , ), we can conclude that < ω arccos( sin ω ω ) < . In a similar way we can prove the other inequality. We can actually verify that when ω ∈ ( , ) arccos(sin ) < c < √ and arccos(sin − sin ) < c < . We now wonder, whether or not f is expandable for ω = .
To answer this question, we have to nd a l( , j) < e . We can easily verify that, for this value of ω, l( , ) ≈ . < e , so we can expand f ∀ω ∈ ( , ]. If we want to expand f for an argument in [− , ), we can just remark that since sin ω = − sin(−ω) for any ω. Furthermore, the following fact is also interesting: consider ω = π ∈ ( , ]; we have sin π = √ . Hence, we can write After some algebraic manipulations, we eventually obtain The next example is about the expansion of composite functions. We can expand these functions in the same way we did before, but we would then obtain for the c j equations that are not solvable with "standard" methods, and so we would have to use approximation methods to nd these values. Hence, it is sometimes better to expand these functions as shown below. We could also expand all the terms with sin x in this equation with the series of Example 6.2, if we wanted.

Approximation of a function with a nite sum and error term
We can now expand a function f according to a new type of series. An important question is: if we consider a nite sum instead of the series, what is the error due to the approximation? To answer this question, we can use the formula What we want to do is to write It is easy to see, looking at these two formulae, that If M (x, i)M (x , i) → when i → +∞, the function can certainly be expanded as a series (the vice versa is not necessarily true, because we have inequalities) as said before.
− ≤ l(x , j) ≤ for large j, which is equivalent to |l(x , j)| ≤ . Therefore, we can expand f when the said conditions are satis ed.
For example, let f (x) = e x , x = , and I = [− , ] (although the precise de nition of I plays no role here). It is easy to verify that the conditions above are satis ed. Therefore, we can write the series, which is obtained simplifying e ,