Behaviour being used: Adaptive mode
Minimum fraction: 0
Question summary: Let us consider [f left parenthesis x right parenthesis] the periodic extension of [# f] and [g left parenthesis x right parenthesis] the periodic extension of [Error converting from MathML to accessible text.] , for [x element of open square brackets negative pi comma pi close square brackets]: [f] is [C to the power of 0] and [C to the power of 1] in [real numbers]. The function [g] is piecewise [C squared]; The equality [# f plus # g equals pi squared over 3 plus 4 sum for n greater-than or slanted equal to 1 of open parentheses negative 1 close parentheses to the power of n over n squared open parentheses cos left parenthesis nx right parenthesis plus fraction numerator 3 sin left parenthesis nx right parenthesis over denominator n end fraction close parentheses] holds for any [x element of open square brackets negative pi comma pi close square brackets]; Uniform convergence can be assured for both [Error converting from MathML to accessible text.] and [SFT left parenthesis g right parenthesis], since the Dirichlet criteria applies and the functions are continuous; The result [sum for n greater-than or slanted equal to 0 of open parentheses negative 1 close parentheses to the power of n over open parentheses 2 n plus 1 close parentheses cubed equals pi cubed over 32] can only be proved using [SFT left parenthesis g right parenthesis]; Employing the Parseval identity, it can be proved that [sum for n greater-than or slanted equal to 1 of # yn equals # sol]
Right answer summary: The equality [# f plus # g equals pi squared over 3 plus 4 sum for n greater-than or slanted equal to 1 of open parentheses negative 1 close parentheses to the power of n over n squared open parentheses cos left parenthesis nx right parenthesis plus fraction numerator 3 sin left parenthesis nx right parenthesis over denominator n end fraction close parentheses] holds for any [x element of open square brackets negative pi comma pi close square brackets]; Uniform convergence can be assured for both [Error converting from MathML to accessible text.] and [SFT left parenthesis g right parenthesis], since the Dirichlet criteria applies and the functions are continuous
Question state: todo