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Question Number 68220 by ~ À ® @ 237 ~ last updated on 07/Sep/19

   Let consider (a_n )_n  and (u_n )_n  two reals  sequence    defined such as   a_0 =1 , ∀ n>1  a_(n+1) =Σ_(p=0) ^n a_p a_(n−p)    and  Σ_(p=0) ^n a_p u_(n−p) =0  Part1  1)Express  ∀ n >1   a_n  in terms of n  2) Find the largest domain of convergence of the integer serie {a_n x^n }  3)Determinate ∀ x∈D the sum f(x) of {a_n x^n }  4)Find the radius of convergence of the serie {u_n x^n }   5) Give the relation that between the sum S(x) of the second serie and (x/(f(x)))   6) Can you developp in integer serie  g(x)=((πx)/(tan(πx)))  Part2  Now do  the part 1   but in the order  2)−1)−3)−4)−5)−6)

$$\:\:\:{Let}\:{consider}\:\left({a}_{{n}} \right)_{{n}} \:{and}\:\left({u}_{{n}} \right)_{{n}} \:{two}\:{reals}\:\:{sequence}\:\: \\ $$ $${defined}\:{such}\:{as}\:\:\:{a}_{\mathrm{0}} =\mathrm{1}\:,\:\forall\:{n}>\mathrm{1}\:\:{a}_{{n}+\mathrm{1}} =\underset{{p}=\mathrm{0}} {\overset{{n}} {\sum}}{a}_{{p}} {a}_{{n}−{p}} \:\:\:{and}\:\:\underset{{p}=\mathrm{0}} {\overset{{n}} {\sum}}{a}_{{p}} {u}_{{n}−{p}} =\mathrm{0} \\ $$ $${Part}\mathrm{1} \\ $$ $$\left.\mathrm{1}\right){Express}\:\:\forall\:{n}\:>\mathrm{1}\:\:\:{a}_{{n}} \:{in}\:{terms}\:{of}\:{n} \\ $$ $$\left.\mathrm{2}\right)\:{Find}\:{the}\:{largest}\:{domain}\:{of}\:{convergence}\:{of}\:{the}\:{integer}\:{serie}\:\left\{{a}_{{n}} {x}^{{n}} \right\} \\ $$ $$\left.\mathrm{3}\right){Determinate}\:\forall\:{x}\in{D}\:{the}\:{sum}\:{f}\left({x}\right)\:{of}\:\left\{{a}_{{n}} {x}^{{n}} \right\} \\ $$ $$\left.\mathrm{4}\right){Find}\:{the}\:{radius}\:{of}\:{convergence}\:{of}\:{the}\:{serie}\:\left\{{u}_{{n}} {x}^{{n}} \right\}\: \\ $$ $$\left.\mathrm{5}\right)\:{Give}\:{the}\:{relation}\:{that}\:{between}\:{the}\:{sum}\:{S}\left({x}\right)\:{of}\:{the}\:{second}\:{serie}\:{and}\:\frac{{x}}{{f}\left({x}\right)}\: \\ $$ $$\left.\mathrm{6}\right)\:{Can}\:{you}\:{developp}\:{in}\:{integer}\:{serie}\:\:{g}\left({x}\right)=\frac{\pi{x}}{{tan}\left(\pi{x}\right)} \\ $$ $${Part}\mathrm{2} \\ $$ $$\left.{N}\left.{o}\left.{w}\left.\:\left.{d}\left.{o}\:\:{the}\:{part}\:\mathrm{1}\:\:\:{but}\:{in}\:{the}\:{order}\:\:\mathrm{2}\right)−\mathrm{1}\right)−\mathrm{3}\right)−\mathrm{4}\right)−\mathrm{5}\right)−\mathrm{6}\right) \\ $$

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