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 an converging to α. The more n increases, the more an 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 nε such that Eqn2.gif for every n>nε.

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 xEqn3.gif

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: Eqn5.gif

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


From (7.3) we have also


Geometric series in WolframAlpha.


Mercator's series.

The fraction Eqn8.gif 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 Eqn9.gif 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


For example, to calculate ln10


Logarithmic series in WolframAlpha.


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 Eqn19.gif we obtain an algorithm to approximate π


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

Arctangent series expansions in WolframAlpha.


MacLaurin's expansions.

If we consider the Mercator's series (7.5), we could find that the coefficients cn of the powers xn in the sum are such that Eqn21.gif, 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 cn 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 ex 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)



Series expansions of the exponential and hyperbolic sine and cosine in WolframAlpha.


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:


Series expansions of the circular sine and cosine in WolframAlpha.


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

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




and finally



Series expansions with WolframAlpha.

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