Series Solutions — Question 1

Question 1 – Solution

Strategy. Follow the surviving coefficient chain and bound its tail by a geometric majorant.

Step 1: Match every coefficient. The coefficient of $x^n$ on the right is $n(n+1)a_n$, so $$\boxed{a_{n+4}=\frac{n\,a_n}{(n+4)(n+3)(n+2)}\quad(n\geq 0).}$$ In particular $a_4=0$, not a freely chosen coefficient. The initial coefficients are $(a_0,a_1,a_2,a_3)=(0,1,0,0)$; only indices $1\pmod 4$ survive.

Step 2: Compute the first terms. Successive use of $n=1,5,9$ gives $$\boxed{y=x+\frac{x^5}{60}+\frac{x^9}{6048} +\frac{x^{13}}{1153152}+\cdots.}$$ The missing coefficients are zero. The first four derivatives at zero match the data, and substitution into the recurrence verifies the displayed chain.

Step 3: Prove convergence and certify the tail. For fixed $x$, successive nonzero terms have ratio in magnitude $|x|^4 n/[(n+4)(n+3)(n+2)]\to 0$, so the series is entire and may be differentiated termwise. For $n\geq 9$, the coefficient ratio is at most $1/(n+2)^2<1/100$. Consequently, for $|x|\leq 1$, $$|y-p_9|\leq\frac{100|x|^{13}}{99\cdot 1153152}\leq B(x):=\frac{11|x|^{13}}{10\cdot 1153152} \leq\frac{11}{10\cdot 1153152}<10^{-6}.$$ This bound concerns the full infinite tail, not just its first omitted term.

Step 4: Check the error’s sign and graph. Every surviving coefficient is positive and every surviving degree is odd. Thus $y-p_9$ is positive for $x>0$, negative for $x<0$, and zero at zero. The figure compares $10^6(y-p_9)$ and $10^6B$ on $[0,1]$.