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" # # # * ( # * ( + # ! # , # ' # 0 ( #( ( ( )i ( ) = # / # # #( 4( ( π )i δ # * # > # ( # )= + # 5( / 6 = ) * 9 # * " # < ∑ ∫ () ) ) π 4( ! # * )i 4( # # # ( # # 9! ) = ∑ 4( ( π +∞ δ B =−∞ ! ' π # # !. &# # , #* . C ( *# # ) #6 4( ## # 4( # # ## # / # ! # ( 4( ( − π ?! = # ( # # 5, * =δ * # # # * # 9# 4( ( π +∞ # # 0 * D $ # %# 0 ( # ( + ! # # # - ( )i ( ) = ; ( ( ( ( # # ( , # ( " # * 0 # 3 !# # ( 0 # # * ! 4( ( π ( − 0 =−∞ + C ∫ A %# # 4( ( − π ( # ' = &# 9 ( 4( # ∫ () * ( ( + . ( . : # ! = @ # 9 ∫ () () + # # ) 9 ( ' , 4( ( π . &# # # > # # 4( !. ; ( ( # # E ! ) !E * # ! # # ( ( # )# * % # / 0 # , *# , ( ( ∫ () ( ! % 9 # + # . ; / + ( ) . & * ; # , 4( #% 0 # ,)# * #( # # * # # 4( # ( )i ( ) = ∑ ∑ ∞ ∞ =−∞ =−∞ 9 &# )i # # 4( # ∫ () ' # π 4( &# ∞ =−∞ ∑ ∞ B δ = =−∞ +∞ ( # # # ) = ∑ ∑ = F! =−∞ # # . ; # 4( ( π ( # ' 9! 4( ( π = = ( ( * # 0 , * ! # ( # # # ## ( #( # * # # * # # # !. # #( # # * !0 # # * # ( ( # * #B #. % & * ! # 4( # + 9! # / − # ( + # #< = ( )= ∑ +∞ = =−∞ 4( ( − π * # # 0 =− − # )= ∑ +∞ − =−∞ ( ( 4( ( π +∞ =−∞ != − ! ( # # )= ( )= ∑ 4( ( π = ! !=− − !! ( # ). = # − ! # < (− ) = " # ( ( ∑ 4( ( − π +∞ =−∞ # # , " # )= ∑ # # +∞ − =−∞ 4( ( π # )=± ( )= ) # ∑± +∞ =−∞ 4( ( π # ). ( . # # * # ' ( )= ∑ 4( ( +∞ =−∞ &# # ) ( * # , ! # )= ∑ , # 3 +∞ !0 ! =−∞ # # !< > # ( # # # * # ( ( #. # G ( , ( # * ( = * ! * 9 ' −τ ) = ∑ = ∫ () * # # 4( ( # , ( # 3 !< ) ∫ () 9 = ' H! ( τ # * 4( − π # τ! # # ( ( !< ( :( )# ! 0 4( ( π +∞ =−∞ # + # # 3 # = ( # , # # ∑ ( −9) ( )= ' ( #( *% 0 # #( ∑ # !# π )= # # * 0 4# #( # ∑ ( ) π # ' . & # ( ( +∞ ). . * # , !. ' 4( ( π ( ). # % , #( . ( I # # # #* ( 0 =−∞ ' ! # =− ) ' 4( ( π +∞ =−∞ ( # 4( ( π +∞ # !( #< 0 #( ( # # # =−∞ ) + 8 !=− # =−∞ &4 # τ ) 4( ( π ) = − ( ) ∑ (− ) * ! 4( ( − π =−∞ 4( ( π +∞ ( ' ∑ +∞ # * # −τ )) = ( 9!. " ( # # 0 0 ( * #0 ( 9! " * # ! , ( # 9!K " # 0 # # # * ! # J( 0 )* # # ( # # < # # , ( + #! . - * %4 ?! " * - * # . * #( # , ! " * * # 0 ( ( # # #( # # # * #/ # # ( ( . # # # # # # ## 0 ( 0 9 →+∞ %# # , # , ! ( >0 , ( + % * # # * / # ) = . !# # ! ( # # L 4 # # FL # < # , #0 # # # / 0 )# # # # ( ( # * , ( . ) ( 0 + + #* # # # # * 4# # . &# , 4( ( π =− ( ## # ∑ ( # # ) + #! ( # #! / # ; # # ( # ' $ ∫ ( )− 9! ( ' ( - : # # * # # # # * / + π # 4( ! # % ( # # * # # # , π 4( # # 0 # !. ! # # . ) * &4 # + # 4( # #0 # # * # # ' #* # . + # δ ! 0 # #)# ( # # # # ( # # 9! : 0 , #< , * 5( ( # # ∑δ( )= +∞ =−∞ − ( >0 # * # # !< ). #( ( , #0 # ( ( # ( ( #( #( ( # # ( # # * # ( #! 6 / > # # # ( = . : ( # # ( # ) ( " # ( ( # ( # # . ! " δ ! # # ( ) # %# 0 ( , > # # ∫δ ( ) ( ) # ( ( 5( , ( ) = !0 ( # , . ( 0 # * # ' 6# = 9 ∫ δ( ) + 4( ( − π ) 4( ( = )= 9 . − , . / $ / # * + * &# ( # ( # # ( ( ( # 3 ## # / # # ' * ( % # # # 9 , 0 ( ( # = =9! # ( / ( # 0 %* # # * # #. ( # ; / ( < # ' # # < 4( ( − π = π ) − # −98 # , ( 0 , # # = C # 0 , # # * # # ! # )= ( ( ∑ +∞ =−∞ =9 # # # / # #( # ∑ +∞ # 9= 98 < ' ( 8π ( # , ?=− ( . ; # ( 9 9 9 π 9 − + − + ... = . ? F H / * 8 ?π! < $ # # / = − + − ... B π ?π Fπ # * . # 3 # % 9 − ( −9) . π π 9 − ( −9) 4( = π 0 &# / , ∑ = = # = 9 ( − 9 98 ! # = +∞ ) ( ( ( 9= 9 − ( −9) π ## 8 Fπ! D 7 , F= 9 − 4( ( − π + π $ < 9 =9 ( # , # # ( 0 . & ( # , / ( α ( # , # ( / # # # , α # # α 0 ## ( # =∑ +∞ 9 =9 # * G =∑ +∞ α= * # # # , # ( ( # # ( ( # # ,# α > ∑ +∞ ⋅ 0 # . " ( 9 =9 ( * / )# −9 / # , ( / ( # * # # # , . =9 ( # * # 9 9 / # ( ( ( # = # # # ∑ +∞ = # ( ( % ( +9 9 α= # - +∞ α =9 ( ) =∑ 9 ( # , # ∑ +∞ * # * # , # *% # % / 0 #9 . # /( ( ( ) # #9# ( ∑ ∑ +∞ 9 =9 ( ## = 9 = # # ( # # # # π +∞ 9 =∑ +∞ # = M π = 9 # , # B −9 * −9 − ( + 9) −9 ( −9) − 9 )= −9 ( − 9) − 9 − 9) −9 ( )+ π . #< = 9 < ≤ . ( # # # # * − 9) = ≤ < # ( 9 * # ( ( )= ( 9 =9 # 0 %# 9 =9 ( %* , ( )= ∑ M π π B M 4 ( # ( = +∞ =−∞ ( " # * * < 9= ; ! # # −9 () ( )− 9. # * # ) < # ( 9 − * ( . " ( # ## 3 ( # - 0 # ( ( ∫( 9 = " # # * , #( − 9) − # ' * ; + ( π 4# ( 9 π ( # ! (9 − ( −9) ) . =− /# 9 , # #< π = / 0 = 8 ( ! ( # # . * ∑ +∞ 9 # # # ( # ∑ = +∞ ( π − 9) =? / 9 = F ? +∞ # / # ∑ +∞ =−∞ ? = = π . N # , # # / / !< ( = ( B −9 # − 9) −9 π + 9 − ( −9) π ( . # 9 − ( −9) 9 − ( −9) − B ? ? π π = 9 ? π # ( 6 , = =9 5( #( = M =∑ # ( ( −9 9 π =9 ( π B N , # = = =−∞ # * #/ ∑ +∞ = = =9 ( ? ( . π ( # 9 = ? :# ( −9 # ( −9 (9 − ( −9) ) − # ( # * 0 # # * − 9) ( ( ; !. " # # # , & − 9) 9 . −9 = / π − #( 9 −9 # = # ( − 9) / 0 # 0 # ## # 3 − 9) ! / C ( = =− ( ( , , * =− # # = N π ? − H π =− M + ? π 9 = 9 − ( −9) 9 − ( −9) + B π π N π ? 9 − B ? ? = π N . # # 9 = H =∑ +∞ ? = ∑ 9 = +∞ ; = =−∞ =9 , −9 # = ? / ( 99 N − ⋅? π ? F + ⋅? π 99 = =∑ +∞ π π = ? ? ⋅F⋅H N F # * 0 ⋅? πM 9 M =9 # ( > %( ) # = # −9 M πM πM = . ⋅ ?? ⋅ F ⋅ H N F ( # %# # % 0 = M , > > 9Hπ M = . 9 9 M N − B F # ( , ⋅? πM # # #E $ 0 E. /$ =1 # # !=−9 # - # ) # #( # − , ' = # # %# #0 # ( ( =− # * ∑ # # ( # # ∑ = +∞ = . ; #( # ( * # ( # # +∞ ( − 9)( π − # 4 # , ( 0 #, # ) # ( # # # # ( ( =9 # ( −9) 9 # # # π #, =9 ( # ( ( − − = ∑ ( −9) , π . =9 ( # # #0 # < > &* # # * %# 0 =−∞ ( , # * > / 0 # ( # #( * −9 = " , / 0 0 * > 0 #* , ( ( # # # ( # ( # # ( ( ( π − 9) O − )O − 9) =− ( π . # # ( =− # #. − 9) O − = π # )O &# . , # ! + 9)( π ( =− + +9 = ∑ ( −9) P +9 P P=9 # # ∑ =9 ( # # # % π = 9 ; = O π + ( = 9 N ( ( = − 9) F F 99 ( 9 − 9) = = 99 ? = = # # # - π F N ⋅ ? ⋅99 ⋅ ?9 ⋅ F #, ? = −9 O π = ? F =∑ * 9 =∑ 9 0 # =9 # N = 9 =9 +∞ % π9 9 O −9 = ( ) 9 π9 9 O = − 9) = 9 O −9 ( π M H 9 . ? − 9) M ⋅ F ⋅ ?9B − 9) ⋅ ? ⋅ N?F + #, # ( = = MO N ⋅ ? ⋅9H F N9π 9 = . ?MF9 MHF 99 # # + B . π9 = . N?FFF F ( )= # # < ∞ ( F F 9 = − − ∑ ( −9) =9 ⋅ ? ⋅ FF + ⋅ ? ⋅ HH − ⋅ ? ⋅ N9B B * +∞ B )O π # 9 ? ⋅ ? ⋅ F ⋅ H ⋅9? ⋅ =∑ − 9) ( ( 9 = F + ∑ ( −9) −9 ( =9 ? F = F − F ⋅ N + ⋅ F ⋅ 9 − ⋅ F ⋅ ? + H ⋅ F ⋅ HN = = N9 B = H? : 0 − = ∑ ( −9) =9 O π ( + − 9) O + − )O = ∑ ( −9) =9 ( )O # = − ⋅ ? + ⋅ ? ⋅99 − 9 P= +9. ## )O # 9 * ( + − 9)O + − P) O # ( 9 ( ( ( P ( P ( P = ∑ ( −9) =9 +∞ ( * π + − 9)O − + − P) O π , #( ; π P > − = &# π P + 9) ( − P P= = ∑ ( −9) ) → # # ! . −9 # ( ( # : RR # 0 *I Q > ≥ ( ( )=− &# 0 = ( =− # ∑ +∞ =9 9 # 9? = ∑ > π 9 , #( ( −9 +∞ ∑ =9 9N = 9 9 9F = , # +∞ ∑ / # +∞ ∑ =9 9 9 = # ( + ( 7 # # # S9T ( −9) ( )O ζ = π # # . 9M 9H = ⋅ MF F9? ⋅ π 9?M ⋅ O # ⋅ ?M H ⋅ π 9M HNM ⋅9MO . / * ( ) 0 9 =ζ ( ). / .Q * 9 S9T! =9 Q . # , =9 ∞ (<88 B π π ⋅ ? 9H ⋅ π 9 F9 ⋅9 O ⋅9H 99 ⋅ π ?? ⋅ O ∑ S9T −9 ( )O #< 2 ( −9) ( )O !. 7 # , −9 # = # =− ( )= −9 ( 9 0 G RR/ < =9 # * # ). , ( )O F ⋅ H ⋅π 9 ⋅9 O +∞ C −9 , # 9 −9 # =−∞ ≠ ( −9) ( )O +∞ ∑=9 ( π) #% F # ( ∑ π) * 4( ( π +∞ O ( ( ( −9) = ( ( ( ( )= " # - % 8 C . # % ( # Revista Escolar de la Olimpíada Iberoamericana de Matemática http://www.campus-oei.org/oim/revistaoim/ Edita: