Question

Let frac{a_{0}}{2}+sum_{n=1}^{infty}(a_{n}cos(nx)+b_{n}sin(nx)) be the Fourier series of f:[-pi,pi] omathbb{R} and frac{a'_{0}}{2}+sum_{n=1}^{infty}(a'_{n}cos(nx)+b'_{n}sin(nx)) be the Fourier series of the derivative f' of f. In the proof of the Uniform Convergence Theorem to show that the Fourier series of f converges absolutely and uniformly we show, in particular, that both series sum_{n=1}^{infty}|a_{n}| and sum_{n=1}^{infty}|b_{n}| converge and then apply the Weierstrass M-test with M_{n}=|a_{n}|+|b_{n}| to show that the series sum_{n=1}^{infty}|a_{n}cos(nx)+b_{n}sin(nx)| converges. the series frac{(a'_{0})^{2}}{2}+sum_{n=1}^{infty}((a'_{n})^{2}+(b'_{n})^{2}) converges applying Bessel's Inequality. a'_{n}=nb_{n} and b'_{n}=-na_{n} using the integration by parts and the assumption that f(-pi)=f(pi). sum_{n=1}^{infty}((a'_{n})^{2}+(b'_{n})^{2}) converges since both series sum_{n=1}^{infty}(a'_{n})^{2} and sum_{n=1}^{infty}(b'_{n})^{2} converge. the sequence {s_{n}}, s_{n}=sum_{k=1}^{n}|a_{k}|, ninmathbb{N}, converges since the series sum_{n=1}^{infty}a_{n} converges. sum_{n=1}^{infty}|a_{n}| and sum_{n=1}^{infty}|b_{n}| converge and lim_{n oinfty}na_{n}=0 and lim_{n oinfty}nb_{n}=0. the sequence {s_{n}}, s_{n}=sum_{k=1}^{n}|a_{k}|, ninmathbb{N}, of the partial sums of the series sum_{n=1}^{infty}|a_{n}| converges since applying the Schwartz inequality 0 le s_{n} le (sum_{k=1}^{infty}(b'_{k})^{2})^{frac{1}{2}}(sum_{k=1}^{infty}frac{1}{k})^{frac{1}{2}}

          Let frac{a_{0}}{2}+sum_{n=1}^{infty}(a_{n}cos(nx)+b_{n}sin(nx)) be the Fourier series of f:[-pi,pi]	omathbb{R} and frac{a'_{0}}{2}+sum_{n=1}^{infty}(a'_{n}cos(nx)+b'_{n}sin(nx)) be the Fourier series of the derivative f' of f.
In the proof of the Uniform Convergence Theorem to show that the Fourier series of f converges absolutely and uniformly we show, in particular, that
both series sum_{n=1}^{infty}|a_{n}| and sum_{n=1}^{infty}|b_{n}| converge and then apply the Weierstrass M-test with M_{n}=|a_{n}|+|b_{n}| to show that the series sum_{n=1}^{infty}|a_{n}cos(nx)+b_{n}sin(nx)| converges.
the series frac{(a'_{0})^{2}}{2}+sum_{n=1}^{infty}((a'_{n})^{2}+(b'_{n})^{2}) converges applying Bessel's Inequality.
a'_{n}=nb_{n} and b'_{n}=-na_{n} using the integration by parts and the assumption that f(-pi)=f(pi).
sum_{n=1}^{infty}((a'_{n})^{2}+(b'_{n})^{2}) converges since both series sum_{n=1}^{infty}(a'_{n})^{2} and sum_{n=1}^{infty}(b'_{n})^{2} converge.
the sequence {s_{n}}, s_{n}=sum_{k=1}^{n}|a_{k}|, ninmathbb{N}, converges since the series sum_{n=1}^{infty}a_{n} converges.
sum_{n=1}^{infty}|a_{n}| and sum_{n=1}^{infty}|b_{n}| converge and lim_{n	oinfty}na_{n}=0 and lim_{n	oinfty}nb_{n}=0.
the sequence {s_{n}}, s_{n}=sum_{k=1}^{n}|a_{k}|, ninmathbb{N}, of the partial sums of the series sum_{n=1}^{infty}|a_{n}| converges since applying the Schwartz inequality
0 le s_{n} le (sum_{k=1}^{infty}(b'_{k})^{2})^{frac{1}{2}}(sum_{k=1}^{infty}frac{1}{k})^{frac{1}{2}}

$Let fraca02+sumn=1^infty(ancos(nx)+bnsin(nx)) be the Fourier series of f:[-pi,pi] omathbbR and fraca'02+sumn=1^infty(a'ncos(nx)+b'nsin(nx)) be the Fourier series of the derivative f' of f. In the proof of the Uniform Convergence Theorem to show that the Fourier series of f converges absolutely and uniformly we show, in particular, that both series sumn=1^infty|an| and sumn=1^infty|bn| converge and then apply the Weierstrass M-test with Mn=|an|+|bn| to show that the series sumn=1^infty|ancos(nx)+bnsin(nx)| converges. the series frac(a'0)^22+sumn=1^infty((a'n)^2+(b'n)^2) converges applying Bessel's Inequality. a'n=nbn and b'n=-nan using the integration by parts and the assumption that f(-pi)=f(pi). sumn=1^infty((a'n)^2+(b'n)^2) converges since both series sumn=1^infty(a'n)^2 and sumn=1^infty(b'n)^2 converge. the sequence sn, sn=sumk=1^n|ak|, ninmathbbN, converges since the series sumn=1^inftyan converges. sumn=1^infty|an| and sumn=1^infty|bn| converge and limn oinftynan=0 and limn oinftynbn=0. the sequence sn, sn=sumk=1^n|ak|, ninmathbbN, of the partial sums of the series sumn=1^infty|an| converges since applying the Schwartz inequality 0 le sn le (sumk=1^infty(b'k)^2)^frac12(sumk=1^inftyfrac1k)^frac12$

Added by Karen B.

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