7. Series expansions

The exponential, circular and hyperbolic functions, whether direct or inverse, like the real constants π and e, would be not much useful if we could not calculate their values.

However, it must be clear that, since they are real numbers, in general it isn't possible to thoroughly express these values, like we do with natural numbers. In general, to calculate a real number α means to find an algorithm which generates a sequence of rational numbers a_n converging to α. The more n increases, the more a_n approximates α. In the practical applications we use sufficiently good approximations.

Maybe the more simple example of such sequences is the sequence of the powers with natural exponent n of a rational number x such that |x|<1. The more n increases, the more the terms xⁿ of this sequence approach 0. We may formally express this by saying that for all rational number x, such that |x|<1

(the limit of the sequence xⁿ equals 0)

if and only if for every positive real number ε there is a natural number n_ε such that for every n>n_ε.

This could be a good starting point to find other useful sequences converging the real numbers generated by the transcendental functions.

Geometric series.

It can easily checked that, for every real number x, 

and that, if x≠1,

The sum in (7.2) is a geometric series, because the sequence of its addends is a geometric sequence, that is the ratio of any two successive addends is constant.

The limit for n→∞ of the series equals the limit of the fraction:

If |x|<1, using a simplified notation,

From (7.3) we have also

Mercator's series.

The fraction is the derivative with respect to x of ln(1+x).

Hence, for |x|<1 and remembering that ln1=0, we can vice versa say that ln(1+x) is the antiderivative of the sum and therefore

So it is possible to approximate the natural logarithm of the numbers in the interval ]0;2[ and the approximation will be better the more one increases the number of the addends in the sum.

The sum (7.5) is said Mercator's series. This series has a limited convergence domain and converges very slowly, but from it we can deduce other series converging over all the logarithm domain.

In fact, from the Mercator's series we have

Subtracting term by term the (7.6) from the (7.5) and remembering the basic properties of the logarithm we obtain

Hyperbolic arctangent.

The (7.7) can be rewritten as

The first term in (7.8) equals the hyperbolic arctangent, so

Circular arctangent.

From the (7.4) we can write

Integrating both the terms and remembering that arctan(0)=0, we have

From the (7.10), remembering that we obtain an algorithm to approximate π

This way to approximate π is known as Leibniz's series.

MacLaurin's expansions.

If we consider the Mercator's series (7.5), we could find that the coefficients c_n of the powers xⁿ in the sum are such that , where f⁽ⁿ⁾(0) denotes the value of the n-th derivative of the function applied to 0 and n! is the factorial of the index n. To can apply the given expression of c_n also when n=0, we assume that f⁽⁰⁾(x) is the function itself and 0!=1.

We could find the same thing for the arctangent functions (7.9) and (7.10) and, in general, for every other infinitely differentiable function f(x), if the function and its derivatives are calculable for x=0. In fact, from

we can get

• 
• 
• 
• 
• 
• …
• 
• …

So for every other infinitely differentiable function f(x), if the function and its derivatives are calculable for x=0, we have

the (7.13) is said MacLaurin's series expansion

The natural exponential and the direct hyperbolic functions.

The more immediate series expansion we can get using the (7.13) is that of the natural exponential e^x for which all the derivatives coincide with the function itself which, when x=0, has value 1.

The series (7.14) allows to approximate the number e=e¹

From the (7.14) we have also

From the (7.14), (7.16) and (3.9), we get the series for the hyperbolic cosine and sine, which could however be directly deduced from the (7.13)

The direct circular functions (cosine and sine).

From the (7.13) we can easily deduce the series expansion of the circular cosine and sine

If we use the (7.17) to expand coshix, where i is the imaginary unit, we get

If we use the (7.18) to expand sinhix, we get

From the (7.21) and the (3.9) we get

From the (7.22) and the (3.9) we get

Gathering all together

These are the well known and very important Euler's formulas.

From the (7.23) we have also

which allows to express a complex number z, with modulus ρ and argument θ, in both the following ways:

The inverse circular functions (arcsine and arccosine).

Given the function

real for x<1, its successive derivatives are

where the double exclamation mark next to the natural number n represents the double factorial of the number itself, that is

• if n is a positive even, n!! is the product between n and all the preceding even numbers;
• if n is a positive odd, n!! is the product between n and all the preceding odd numbers;
• if n=0 or n=-1, n!! = 1.

Since f(0)=1, the MacLaurin's series expansion of f(x) results

Moreover, for -1<x<1, we have

and , finally

The cosine of an angle is equal to the sine of its complementary angle, therefore

We can obtain the expansion (7.24) in a more direct way if we expand the function (7.23) using the generalized binomial coefficients. Infact from

we have

The inverse hyperbolic functions (arcsine and arccosine).

In a similar way we can get

The hyperbolic arccosine is real only if its argument is ≥1. If, however, we want its MacLaurin series expansion, we can proceed as follows

Furthermore we have

so

and finally

Other very useful series expansions can be found in Fourier series expansion.