The question is how to prove this equality ∑ n = 2 k 1 n = ∑ n = 2 ∞ ( 1 n − 1 n + k − 1 )

Question

The question is how to prove this equality      ∑          n      =      2              k            1    n    =      ∑          n      =      2              ∞            (          1      n        −          1              n        +        k        −        1              )

Dustin Durham · Accepted Answer

Write out some terms of the righthand side:                                    (                      1            2                    −                      1                          k              +              1                                )                +                  (                      1            3                    −                      1                          k              +              2                                )                +        …        +                  (                      1            k                    −                      1                          2              k              −              1                                )                                                  +                  (                      1                          k              +              1                                −                      1                          2              k                                )                +                  (                      1                          k              +              2                                −                      1                          2              k              +              1                                )                +        …            Notice that all of the negative terms are cancelled out by positive terms later in the series:   −      1          k      +      1       in the first term by       1          k      +      1       in the k-th term,   −      1          k      +      2       in the second term by       1          k      +      2       in the   (  k  +  1  )-st term, and so on. The only terms that are left uncancelled are the positive parts of the terms in the top line above:      1    2    +      1    3    +  …  +      1    k      .This is precisely the sum on the lefthand side.

Petrovcic2x · Accepted Answer

We may also try this:      ∑          n      =      2              k            1    n    =      ∑          n      =      2              ∞            (          1      n        −          1              n        +        k        −        1              )    =      ∑          n      =      2              ∞            1    n    −      ∑          n      =      2              ∞            1          n      +      k      −      1            ∑          n      =      2              ∞            1          n      +      k      −      1        =      ∑          n      =      2              ∞            1    n    −      ∑          n      =      2              k            1    n    =      ∑          n      =      k      +      1              ∞            1    n    .Q.E.D.

The question is how to prove this equality <munderover> ∑ <mrow class="

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New Questions in Calculus 2