What is the limit to infinity of this function with summation? I need someone with knowledge of limits to infinity and limits of summations to please work out the following: lim_(i->oo)(2^(i)+sum_(j=0)^(i-1) (-1)^j2^j)/(2^(i+1))

easternerjx

easternerjx

Answered question

2022-09-24

What is the limit to infinity of this function with summation?
I need someone with knowledge of limits to infinity and limits of summations to please work out the following:
lim i 2 i + j = 0 i 1 ( 1 ) j 2 j 2 i + 1
For context, I want to determine if the following sequence approaches 2 3 :
1 2 , 3 4 , 5 8 , 11 16 , 21 32 , 43 64 , 85 128 , 171 256 , 341 512 , . . .
Thank you =)

Answer & Explanation

Alec Reid

Alec Reid

Beginner2022-09-25Added 9 answers

Your sequence does not converge. Firstly j = 0 i 1 ( 1 ) j 2 j is sum of geometric series with quotient -2 and is equal to 1 3 ( 1 ( 2 ) i )
We have
lim i 2 i + j = 0 i 1 ( 1 ) j 2 j 2 i + 1 = lim i 2 i 2 i + 1 + lim i 1 ( 2 ) i 3 × 2 i + 1 = 1 2 + lim i 1 3 × 2 i + 1 + lim i ( 2 ) i 3 × 2 i + 1 = 1 2 1 6 lim i ( 1 ) i
Since the last does not converge, the limit does not converge.
Actually the sequence with i th term a i = 1 2 1 6 ( 1 ) i has two subsequences, that converge to 1 / 3 and 2 / 3, respectively:
lim i a 2 i = 2 3 , lim i a 2 i 1 = 1 3
Luisottifp

Luisottifp

Beginner2022-09-26Added 1 answers

j = 0 i 1 ( 1 ) j 2 j = 1 + 2 4 + . . . = 1 ( 1 ( 2 ) i 1 + 1 ) 1 ( 2 ) = ( 2 ) i 1 3
lim i 2 i 1 + j = 0 i 1 ( 1 ) j 2 j 2 i + 1 = lim i 2 i 1 + ( 2 ) i 1 3 2 i + 1

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