Monday, 18 March 2019

sequences and series - Compute 1cdotfrac12+2cdotfrac14+3cdotfrac18+cdots+ncdotfrac12n+cdots




I have tried to compute the first few terms to try to find a pattern but I got



12+12+38+416+532+664



but I still don't see any obvious pattern(s). I also tried to look for a pattern in the question, but I cannot see any pattern (possibly because I'm overthinking it?) Please help me with this problem.


Answer



I=12+24+38+416+532+664+
2I=1+1+34+48+516+632+
2II=1+(112)+(3424)+(4838)+(516416)+

I=1+12+14+18+=2


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