Appendix A
Taylor Series
Consider a real-valued function f (t) whose first n derivatives are continuous on an
interval [a, b] and its derivative of order (n + 1) exists on this interval. Then, for
each t in [a, b], there is c between a and t such that
f (t) = f (a) +
n
r=1
(t − a) r
r!
f
(r) (a) + R n (c, t),
(A.1)
where
R n (c, t) =
(t − a) n+1
(n + 1)!
f
(n+1) (c)
(A.2)
is the remainder of the representation of f (t), and f (r) (a) denotes the rth derivative
d r f (t)/dt r of f (t) at t = a [2] (Theorem 21.1).
This implies that the error of truncated series
f n (t) = f (a) +
n
r=1
(t − a) r
r!
f
(r) (a)
(A.3)
is of the order R n (c, t), i.e., order O
(t − a) n+1
. We use this observation to
construct finite difference representation of derivatives and other approximations.
The Taylor theorem can be used to show the validity of the L’Hôpital rule. Suppose that (1) the ratio f 1 (t)/f 2 (t) of the functions f 1 (t) and f 2 (t) is indeterminate
at t = t 0 , e.g., f 1 (t 0 ) = f 2 (t 0 ) = 0, which results in 0/0, (2) the functions f 1 and
f 2 are differentiable, and (3) the derivative of f 2 is not zero at t 0 , i.e., f
2 (t 0 ) = 0.
The Taylor series expansions of these functions around t 0 give
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Switzerland AG 2021
M. D. Grigoriu, Linear Dynamical Systems,
https://doi.org/10.1007/978-3-030-64552-6
133
Taylor Series
Consider a real-valued function f (t) whose first n derivatives are continuous on an
interval [a, b] and its derivative of order (n + 1) exists on this interval. Then, for
each t in [a, b], there is c between a and t such that
f (t) = f (a) +
n
r=1
(t − a) r
r!
f
(r) (a) + R n (c, t),
(A.1)
where
R n (c, t) =
(t − a) n+1
(n + 1)!
f
(n+1) (c)
(A.2)
is the remainder of the representation of f (t), and f (r) (a) denotes the rth derivative
d r f (t)/dt r of f (t) at t = a [2] (Theorem 21.1).
This implies that the error of truncated series
f n (t) = f (a) +
n
r=1
(t − a) r
r!
f
(r) (a)
(A.3)
is of the order R n (c, t), i.e., order O
(t − a) n+1
. We use this observation to
construct finite difference representation of derivatives and other approximations.
The Taylor theorem can be used to show the validity of the L’Hôpital rule. Suppose that (1) the ratio f 1 (t)/f 2 (t) of the functions f 1 (t) and f 2 (t) is indeterminate
at t = t 0 , e.g., f 1 (t 0 ) = f 2 (t 0 ) = 0, which results in 0/0, (2) the functions f 1 and
f 2 are differentiable, and (3) the derivative of f 2 is not zero at t 0 , i.e., f
2 (t 0 ) = 0.
The Taylor series expansions of these functions around t 0 give
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Switzerland AG 2021
M. D. Grigoriu, Linear Dynamical Systems,
https://doi.org/10.1007/978-3-030-64552-6
133