How to find the value of ∑ k = 1 ∞ ( 1 9 ) k using partial sums?So I was trying to prove an infinite sum by looking at the partial sum, when I ran into a problem.Consider: ∑ k = 1 n ( 1 9 ) k = 1 8 9 − n ( 9 n − 1 )but as there are no solutions to 9 − n ( 9 n − 1 ) = 1, is it possible to prove that ∑ k = 1 ∞ ( 1 9 ) k = 1 8 by the partial sum?

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How to find the value of       ∑          k      =      1        ∞    (      1    9        )    k   using partial sums?So I was trying to prove an infinite sum by looking at the partial sum, when I ran into a problem.Consider:      ∑          k      =      1        n              (              1        9            )        k    =      1    8        9          −      n        (      9    n    −  1  )but as there are no solutions to       9          −      n        (      9    n    −  1  )  =  1, is it possible to prove that      ∑          k      =      1        ∞              (              1        9            )        k    =      1    8  by the partial sum?

Alisa Jacobs · Accepted Answer

You&#039;re right there: You want       lim          n      →      ∞            1    8    (      9          −      n        (      9    n    −  1  )  )Distribute:       lim          n      →      ∞            1    8    (  1  −      9          −      n        )  =      1    8   since       9          −      n        →  0 as   n  →  ∞The problem you say that you &quot;ran into&quot; seems to be more of a conceptual issue. The problem I believe you&#039;re having is that there is no solution to        9          −      n        (      9    n    −  1  )  =  1. This is true, but limits are not exact solutions always. Limits are what a sequence approaches. This means that we can get as close as we want, but that doesn&#039;t mean that we ever have to get there exactly. Thus, while there are no solutions to       9          −      n        (      9    n    −  1  )  =  1 as   n  →  ∞ the left hand side approaches 1, and that&#039;s good enough.

spockmonkey40 · Accepted Answer

∑                      k            =            1                                ∞                                                (                          1              9                        )                    k                                    =                  lim                      n            →            ∞                                    ∑                      k            =            1                                n                                                (                          1              9                        )                    k                                                  =                  lim                      n            →            ∞                                    1          8                          9                      −            n                          (                  9          n                −        1        )                                          =                  1          8                          lim                      n            →            ∞                          (        1        −                  9                      −            n                          )                                          =                  1          8                (        1        −        0        )                                          =                  1          8

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