Let - n- be a positive integer and define - f-n-1-2-3-n-Find Polynomials - P-x- and - Q-x- such that - f-n-2-Q-n-f-n-P-n-f-n-1- forall - n- 1-Find P-2-

f(n)=1!+2!+3!+⋯ +n! f(n+2)=Q(n)f(n)+P(n)f(n+1) 1!+2!+⋯ +(n+2)!=Q(n)[1!+2!+⋯ +n!]+P(n)[1!+2!+⋯ +(n+1)!] Comparing coefficients upto (n 1)! ...

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Coefficients of a Polynomial

Let n b...

Question

Let $n$ be a positive integer and define $f (n) = 1! + 2! + 3! + \dots + n!$ .Find Polynomials $P (x)$ and $Q (x)$ such that $f (n + 2) = Q (n) f (n) + P (n) f (n + 1)$ forall $n \geq 1$
Find P(2).

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