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Question Number 122853 by rs4089 last updated on 20/Nov/20

Commented by Dwaipayan Shikari last updated on 20/Nov/20

y △y △^2 y 2 3 5 2 5 10 2 7 17 2 9 26 φ(y)=2+3(n−1)+(n−1)(n−2)=2+n^2 −1=n^2 +1 Σ^∞ (−1)^n (1/(n^2 +1))=(1/2)−(1/5)+(1/(10))−...=S Σ_(n=1) ^∞ (1/(n^2 +1))=((π+1)/2)+(π/(e^(2π) −1))=(1/2)+(1/5)+(1/(10))+...=S′ S+S^′ =2((1/2)+(1/(10))+(1/(26))+...) S=2((1/2)+(1/(10))+(1/(26))+...)−((π+1)/2)−(π/(e^(2π) −1))=0.636017644

$${y}\:\:\:\bigtriangleup{y}\:\bigtriangleup^{\mathrm{2}} {y} \\ $$$$\mathrm{2} \\ $$$$\:\:\:\:\:\:\mathrm{3} \\ $$$$\mathrm{5}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2} \\ $$$$\:\:\:\:\:\:\mathrm{5} \\ $$$$\mathrm{10}\:\:\:\:\:\:\:\:\:\:\mathrm{2} \\ $$$$\:\:\:\:\:\:\mathrm{7} \\ $$$$\mathrm{17}\:\:\:\:\:\:\:\mathrm{2} \\ $$$$\:\:\:\:\:\:\mathrm{9} \\ $$$$\mathrm{26} \\ $$$$\phi\left({y}\right)=\mathrm{2}+\mathrm{3}\left({n}−\mathrm{1}\right)+\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)=\mathrm{2}+{n}^{\mathrm{2}} −\mathrm{1}={n}^{\mathrm{2}} +\mathrm{1} \\ $$$$\overset{\infty} {\sum}\left(−\mathrm{1}\right)^{{n}} \frac{\mathrm{1}}{{n}^{\mathrm{2}} +\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{5}}+\frac{\mathrm{1}}{\mathrm{10}}−...={S} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} +\mathrm{1}}=\frac{\pi+\mathrm{1}}{\mathrm{2}}+\frac{\pi}{{e}^{\mathrm{2}\pi} −\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{5}}+\frac{\mathrm{1}}{\mathrm{10}}+...={S}' \\ $$$${S}+{S}^{'} =\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{10}}+\frac{\mathrm{1}}{\mathrm{26}}+...\right) \\ $$$${S}=\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{10}}+\frac{\mathrm{1}}{\mathrm{26}}+...\right)−\frac{\pi+\mathrm{1}}{\mathrm{2}}−\frac{\pi}{{e}^{\mathrm{2}\pi} −\mathrm{1}}=\mathrm{0}.\mathrm{636017644} \\ $$$$ \\ $$

Commented by rs4089 last updated on 20/Nov/20

how we can use fourier series of e^(ax) ?

$${how}\:{we}\:{can}\:{use}\:{fourier}\:{series}\:{of}\:\:{e}^{{ax}} \:\:? \\ $$

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