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Question Number 190622 by mnjuly1970 last updated on 07/Apr/23
prove:∫−∞∞(x⋮―)2dx=∑∞k=11k2⋖
Answered by leodera last updated on 17/May/23
cosh2(x)=(ex+e−x2)2=e2x4(1+e−2x)2∫−∞∞x2cosh2(x)dx=8∫0∞x2e−2x(1+e−2x)2dx=8∫0∞x2e−2x∑∞k=0e−2kx(−1)k(1+k)dx=8∑∞k=0(−1)k(1+k)∫0∞x2e−(2+2k)xdx=16∑∞k=0(−1)k(1+k)23(1+k)3=2∑∞k=0(−1)k1(1+k)22∑∞k=1(−1)k−11k2=2η(2)=2(1−21−2)ζ(2)=ζ(2)=∑∞k=11k2=π26
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