Rational Functions

The product of two formal power series
MATH
and
MATH
is the formal power series $A(x)$ given by
MATH
where the coefficients $a_k$ satisfy
MATH
for $k=0,1,2,\cdots .$

If the leading coefficient $b_0$ of $B(x)$ equal 1, then the above formula gives
MATH
Thus if the formal power series
MATH
satisfies
MATH
then its coefficients MATH can be computed with formula (), with the understanding $b_k=0$ for $k>s$. For instance, if $s=4$, $r=2$, then
MATH

MATH

MATH

MATH

MATH

MATH

MATH

MATH

MATH

MATH

MATH
We shall show that if a formal power series $P(x)=\sum p_kx^k$ ``converges'' on an interval, then the resulting function has derivatives of all orders and its coefficients $p_k$ are given by, exactly like the polynomials,
MATH
More is true: if $P(x)$ takes the form $\sum p_n(x-a)^n$ then MATH for MATH. This fact suggests a computational procedure to find $r^{(n)}(a)$ for a rational function $r:$ first, express $r(x)$ in the form MATH with MATH and MATH. Notice that $d_0 $ cannot be zero, or else $p(a)$ would not be defined. Once $d_0$ is factored out from $q(x)$, $r(x)$ takes the form
MATH

Now the above method of expansion is applicable. Once the expansion
MATH
is found $r^{(k)}(a)$ is calculated from the formula $r^{(k)}(a)=c_kk!.$

Example: Compute $c_{k}$ for $k=0,1,2,\cdots ,10$ in the expansion
MATH

MATH

MATH

Exercise:

(a) Find the power series expansion of MATH up to $x^{30}.$

$(b)$ Let MATH. Find $f^{(n)}(0)$ for MATH

N-th Order Derivatives of Particular Rational Functions

The expansion
MATH
holds (the geometric series). Therefore, if $f(x)=\frac 1{1-x}$ then $f^{(n)}(0)=n!$ for all $n.$

Replacing $x$ by $ax$, we have the expansion
MATH
Therefore, if $f(x)=\frac 1{1-ax}$ then $f^{(n)}(0)=a^nn!$ for all $n$. To find the expansion for functions of the form $\frac 1{a-x}$ we write it in the form MATH to obtain the expansion
MATH
Therefore, if $f(x)=\frac 1{a-x}$ then MATH. By differentiating the geometric series on both sides, we obtain
MATH
Therefore, if MATH then $f^{(n)}(0)=(n+1)n!$. Again, differentiating this series we obtain
MATH
Therefore, if MATH then MATH

Exercise:

Find out $f^{(n)}(0)$ if MATH. What if MATH What if MATH What if MATH

If the rational function has the partial fractions decomposition in the form
MATH
the above method may be applied to find $r^{(n)}(0).$

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