Search results
Results from the WOW.Com Content Network
A power series = is convergent for some values of the variable x, which will always include x = c since () = and the sum of the series is thus for x = c. The series may diverge for other values of x , possibly all of them.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
The power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any unital Banach algebra B. In this setting, e 0 = 1, and e x is invertible with inverse e −x for any x in B. If xy = yx, then e x + y = e x e y, but this identity can fail ...
The six most common definitions of the exponential function = for real values are as follows.. Product limit. Define by the limit: = (+).; Power series. Define e x as the value of the infinite series = =! = + +! +! +! + (Here n! denotes the factorial of n.
The original proof is based on the Taylor series expansions of the exponential function e z (where z is a complex number) and of sin x and cos x for real numbers x . In fact, the same proof shows that Euler's formula is even valid for all complex numbers x .
A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms.Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value).
A power series is a series of the form = (). The Taylor series at a point of a function is a power series that, in many cases, converges to the function in a neighborhood of . For example, the series
Alternatively, the equality can be justified by multiplying the power series on the left by 1 − x, and checking that the result is the constant power series 1 (in other words, that all coefficients except the one of x 0 are equal to 0). Moreover, there can be no other power series with this property.