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z{e2|7c!B*~%G|}(g-$@=!AgNaU_{sykcZ&ki~U^L#NOP64w;JRn*s^|cxut6fgc{B z@$vA35y^`{+JC=lSOi)iePdVxN}}=cLcj>?C15!@n2!&+v41bRw{+Nlzk{Lt0#F_V z_!p3goQGH7A9o=Fd;$o2=(uddU!6f3Y6~wKM)A{hk<`${E&ZQFxUgtKp+@D?_bs6NUr~p@-O&M z-oKc@KavGNJpX7(fLGvwFd&csKhHnN2tfW@z`y4s2LQwadf*`v`X(R@NLmtQl z=0n=~upl1<0(+nWay7vZX9kJ!!XD;ALI1EA34_Lk;|JJ;6|_#ei>K>rv*Fbs@j z_`nhv4=?y1xxBm}_(MOD4>Q<9-(i18{b4~q{(sm6byDJeqt{~uyNhkQHh;%egT>gjB5fenU1pfE5tGdi=h IiVXJu2Xv7{r2qf` diff --git a/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations.tex b/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations.tex index 9e15b12..102dc1e 100644 --- a/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations.tex +++ b/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations.tex @@ -30,4 +30,5 @@ &+ \coeffA{m}{8} \left[ \frac{1}{218790} (-1551693 n + 2042040 n^3 - 516868 n^5 + 26520 n^7 + n^{17}) \right] + \cdots \end{split}\label{eq:equation17} \end{equation} -\input{coefficients_via_system_of_equations_example1} \ No newline at end of file +\input{coefficients_via_system_of_equations_example1} +\input{coefficients_via_system_of_equations_example2} \ No newline at end of file diff --git a/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations_example2.tex b/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations_example2.tex index 3994ba2..391483b 100644 --- a/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations_example2.tex +++ b/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations_example2.tex @@ -1,36 +1,36 @@ -For $m=2$ we have an identity - +\begin{examp} + Let be fixed $m=2$ so that we have the following relation defined by (eqref) + \begin{equation*} + \coeffA{m}{0} n + + \coeffA{m}{1} \left[ \frac{1}{6} (-n + n^3) \right] + + \coeffA{m}{2} \left[ \frac{1}{30} (-n + n^5) \right] - n^5 = 0 + \end{equation*} +\end{examp} +Multiplying by $30$ both, right hand side and left hand side, we get \begin{equation*} - \coeffA{m}{0} n - + \coeffA{m}{1} \left[ \frac{1}{6} (-n + n^3) \right] - + \coeffA{m}{2} \left[ \frac{1}{30} (-n + n^5) \right] - n^5 = 0 + 30 \coeffA{2}{0} n + 5 \coeffA{2}{1} (-n + n^3) + \coeffA{2}{2} (-n + n^5) - 30n^5 = 0 \end{equation*} -Multiplying by 30 both parts we get +Opening brackets and rearranging the terms gives \begin{equation*} - 30 \coeffA{m}{0} n - + 5 \coeffA{m}{1} (-n + n^3) - + \coeffA{m}{2} (-n + n^5) - 30 n^5 = 0 + 30 \coeffA{2}{0} - 5 \coeffA{2}{1} n + 5 \coeffA{2}{1} n^3 - \coeffA{2}{2} n + \coeffA{2}{2} n^5 - 30n^5 = 0 \end{equation*} - -Opening brackets gives +Combining the terms yields \begin{equation*} - 30 \coeffA{m}{0} n -5 \coeffA{m}{1} n + 5 \coeffA{m}{1} n^3 -\coeffA{m}{2} n + n^5 \coeffA{m}{2} n^5 - 30 n^5 = 0 + n (30 \coeffA{2}{0} - 5 \coeffA{2}{1} - \coeffA{2}{2}) + 5 \coeffA{2}{1} n^3 + n^5 (\coeffA{2}{2} - 30) = 0 \end{equation*} - -Arranging terms we get - +Therefore, the system of linear equations follows \begin{equation*} - n (30 \coeffA{m}{0} n - 5 \coeffA{m}{1} -\coeffA{m}{2}) + 5 \coeffA{m}{1} n^3 + n^5 (\coeffA{m}{2} n^5 - 30) = 0, \quad n \geq 1 + \begin{cases} + 30 \coeffA{2}{0} - 5 \coeffA{2}{1} - \coeffA{2}{2} = 0 \\ + \coeffA{2}{1} = 0 \\ + \coeffA{2}{2} - 30 = 0 + \end{cases} \end{equation*} - -Hence - +Solving it we get \begin{equation*} \begin{cases} - \coeffA{m}{1} = 0 \\ - \coeffA{m}{2} n^5 - 30 = 0 \\ - 30 \coeffA{m}{0} n - 5 \coeffA{m}{1} -\coeffA{m}{2} = 0 + \coeffA{2}{2} = 30 \\ + \coeffA{2}{1} = 0 \\ + \coeffA{2}{0} = 1 \end{cases} -\end{equation*} - -So that $\coeffA{m}{1}=6 \quad \coeffA{m}{1} = 1$ \ No newline at end of file +\end{equation*} \ No newline at end of file diff --git a/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations_example3.tex b/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations_example3.tex index e69de29..5e35c0e 100644 --- a/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations_example3.tex +++ b/src/sections/02_coefficients_via_system_of_equations/coefficients_via_system_of_equations_example3.tex @@ -0,0 +1,37 @@ +\begin{examp} + Let be fixed $m=3$ so that we have the following relation defined by (eqref) + \begin{equation*} + \coeffA{m}{0} n + + \coeffA{m}{1} \left[ \frac{1}{6} (-n + n^3) \right] + + \coeffA{m}{2} \left[ \frac{1}{30} (-n + n^5) \right] + + \coeffA{m}{3} \left[ \frac{1}{420} (-10 n + 7 n^3 + 3 n^7) \right] - n^7 = 0 + \end{equation*} +\end{examp} +Multiplying by $30$ both, right hand side and left hand side, we get +\begin{equation*} + 30 \coeffA{2}{0} n + 5 \coeffA{2}{1} (-n + n^3) + \coeffA{2}{2} (-n + n^5) - 30n^5 = 0 +\end{equation*} +Opening brackets and rearranging the terms gives +\begin{equation*} + 30 \coeffA{2}{0} - 5 \coeffA{2}{1} n + 5 \coeffA{2}{1} n^3 - \coeffA{2}{2} n + \coeffA{2}{2} n^5 - 30n^5 = 0 +\end{equation*} +Combining the terms yields +\begin{equation*} + n (30 \coeffA{2}{0} - 5 \coeffA{2}{1} - \coeffA{2}{2}) + 5 \coeffA{2}{1} n^3 + n^5 (\coeffA{2}{2} - 30) = 0 +\end{equation*} +Therefore, the system of linear equations follows +\begin{equation*} + \begin{cases} + 30 \coeffA{2}{0} - 5 \coeffA{2}{1} - \coeffA{2}{2} = 0 \\ + \coeffA{2}{1} = 0 \\ + \coeffA{2}{2} - 30 = 0 + \end{cases} +\end{equation*} +Solving it we get +\begin{equation*} + \begin{cases} + \coeffA{2}{2} = 30 \\ + \coeffA{2}{1} = 0 \\ + \coeffA{2}{0} = 1 + \end{cases} +\end{equation*} \ No newline at end of file