Power series
Maclaurin and Taylor series
The power series expansion of the hyperbolic sine and hyperbolic cosine function
Properties of the power series expansion of the hyperbolic sine and hyperbolic cosine function
Maclaurin and Taylor series
Consider the polynomial function
f (x) = an xn + an - 1 xn - 1 + · · · a3 x3 + a2 x2 + a1 x + a0.
If we write the value of the function and the values of its successive derivatives, at the origin, then
f (0) = a0,     f '(0) = 1· a1,     f ''(0) = 1· 2a2,     f '''(0) = 1· 2· 3a3,  . . .  ,  f (n)(0) = n!an
 so we get the coefficients;
Then, the polynomial  f (x) with infinitely many terms, written as the power series
 and
where 0! = 1,   f (0) (x0) =  f (x0) and  f (n) (x0) is the nth derivative of  f at x0,
represents an infinitely differentiable function and is called Maclaurin series and Taylor series respectively.
The power series expansion of the hyperbolic sine and hyperbolic cosine function
We use sum and difference of two convergent series
to represent the hyperbolic sine (sinh or sh) and hyperbolic cosine (cosh or ch) function by the power series.
 Since,
 then,
 and
Properties of the power series expansion of the hyperbolic sine and hyperbolic cosine function
 The hyperbolic sine
The hyperbolic sine is represented by the infinite polynomial lacking every next even term so that  an-1 = 0
 therefore, the coordinates of translations,
Thus, the infinite source odd polynomials describe the graph of the hyperbolic sine,
 The hyperbolic cosine
The hyperbolic cosine is represented by the infinite polynomial lacking every next odd term so that  an-1 = 0
 therefore, the coordinates of translations,
Thus, the infinite source even polynomials translated by  y0 = 1 describe the graph of the hyperbolic cosine,
Calculus contents B