How can I evaluate this?
∞∑n=11n(n+2)(n+4)=11⋅3⋅5+12⋅4⋅6+13⋅5⋅7+14⋅6⋅8+⋯
I have tried:
11⋅3⋅5+13⋅5⋅7+12⋅4⋅6+14⋅6⋅8+⋯=13⋅5(1+17)+14⋅6(12+18)+⋯
and so on... Been stuck for a while. Result should be 1196
Answer
With *partial fractions:1k(k+2)(k+4)=18(1k−2k+2+1k+4), we get a telescoping sum which simplifies to: 18(1+12−13−14−1n+1−1n+2+1n+3+1n+4).
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