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 =)

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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 =)

Alec Reid · Accepted Answer

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 · Accepted Answer

∑          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

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))

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