
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) = a_{n
}x^{n}
+ a_{n}_{ }_{}_{
1 }x^{n }^{}^{
}^{1} + · · · + a_{3 }x^{3}
+ a_{2 }x^{2} + a_{1 }x
+ a_{0}.

If
we write the value of the function and the values of its successive
derivatives, at the origin, then 
f
(0) =
a_{0}, f '(0) = 1· a_{1},
f ''(0) = 1· 2a_{2},
f '''(0) = 1· 2· 3a_{3}, . . . , f ^{(}^{n}^{)}(0) =
n!a_{n} 
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) }(x_{0})
= f
(x_{0})
and
f ^{(}^{n}^{)
}(x_{0})
is the nth
derivative of f at
x_{0}, 
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 a_{n}_{}_{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 a_{n}_{}_{1}
= 0 
therefore, the
coordinates of translations, 




Thus,
the infinite source even polynomials translated by y_{0}
= 1 describe the graph of the hyperbolic
cosine, 










Calculus
contents B 



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