Could you please explain how come: sum_(i=1)^infty (0.5)^(i+1)(i+1)=2

Yaretzi Mcconnell 2022-11-20 Answered
Could you please explain how come:
i = 1 ( 0.5 ) i + 1 ( i + 1 ) = 2
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Answers (1)

Kennedy Evans
Answered 2022-11-21 Author has 16 answers
We know for | r | < 1
1 i < r i + 1 = r 2 1 1 r = r 2 1 r
Now,  d ( 1 i < r i + 1 ) d r = 1 i < d r i + 1 d r = 1 i < ( i + 1 ) r i
and  d ( r 2 1 r ) d r = 2 r r 2 ( 1 r ) 2
1 i < ( i + 1 ) r i = 2 r r 2 ( 1 r ) 2
1 i < ( i + 1 ) r i + 1 = r ( 2 r r 2 ) ( 1 r ) 2
Alternatively, using Arithmetico-geometric series, we can show,
1 i < ( i + c ) r i + 1 = ( c + 1 ) r 2 c r 3 ( 1 r ) 2
where c is any finite number
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