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

Eqn1.gif

(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,

Eqn4.gif

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,

Eqn6.gif

From (7.3) we have also

Eqn7.gif

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

Eqn10.gif

Eqn11.gif

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

Eqn12.gif

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

Eqn13.gif

For example, to calculate ln10

Eqn14.gif

Logarithmic series in WolframAlpha.

 


Hyperbolic arctangent.

The (7.7) can be rewritten as

Eqn15.gif

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

Eqn16.gif

 


Circular arctangent.

From the (7.4) we can write

Eqn17.gif

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

Eqn18.gif

From the (7.10), remembering that Eqn19.gif we obtain an algorithm to approximate π

Eqn20.gif

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

Eqn22.gif

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

Eqn29.gif

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.

Eqn30.gif

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

Eqn31.gif

From the (7.14) we have also

Eqn32.gif

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)

Eqn33.gif

Eqn34.gif

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

Eqn35.gif

Eqn36.gif

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

Eqn37.gif

Eqn38.gif

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

Eqn39.gif

Eqn40.gif

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

Eqn41.gif

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

Eqn42.gif

Gathering all together

Eqn43.gif

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

From the (7.23) we have also

Eqn44.gif

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

Eqn45.gif

Series expansions of the circular sine and cosine in WolframAlpha.

 


The inverse circular functions (arcsine and arccosine).

Given the function

Eqn047.gif

real for x<1, its successive derivatives are

Eqn048.gif

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

Eqn049.gif

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

Eqn050.gif

Eqn051.gif

and , finally

Eqn052.gif

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

Eqn053.gif

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

Eqn054.gif

we have

Eqn055.gif

 


The inverse hyperbolic functions (arcsine and arccosine).

In a similar way we can get

Eqn056.gif

Eqn057.gif

Eqn058.gif

 

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

Eqn059.gif

Furthermore we have

Eqn060.gif

so

Eqn061.gif

and finally

Eqn062.gif

 

Series expansions with WolframAlpha.


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