How does one sum the series S=a−23a3+2⋅43⋅5a5−2⋅4⋅63⋅5⋅7a7+⋯
This was asked to me by a high school student, and I am embarrassed that I couldn't solve it. Can anyone give me a hint?!
Answer
HINT (a2+1)S′=1−aS by transmuting the coefficient recurrence to a differential equation.
⇒1=(a2+1)S′+aS=f(fS)′ for f=(a2+1)1/2
⇒S=f−1∫f−1=sinh−1(a)(a2+1)1/2
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