How to prove ∑ i = 1 n − 1 1 lcm ⁡ ( a i , a i + 1 ) &lt; 1 where a i ∈ N and a i &lt; a i + 1 ?

Question

How to prove       ∑          i      =      1              n      −      1            1          lcm      ⁡      (              a        i            ,              a                  i          +          1                    )        &amp;lt;  1 where       a    i    ∈      N   and       a    i    &amp;lt;      a          i      +      1       ?

Melanie Wong · Accepted Answer

Notice that  S  :=      ∑          i      =      1              ∞            1          lcm              (                  a                      i                          ,                  a                      i            +            1                          )              =      ∑          i      =      1              ∞                  gcd      (              a                  i                    ,              a                  i          +          1                    )                      a                  i                            a                  i          +          1                      =      ∑          i      =      1              ∞                  gcd      (              a                  i                    ,              a                  i          +          1                    )                      a                  i          +          1                    −              a                  i                          (          1              a                  i                      −          1              a                  i          +          1                      )    .Since   gcd  (      a          i        ,      a          i      +      1        )  =  gcd  (      a          i        ,      a          i      +      1        −      a          i        )  ≤      a          i      +      1        −      a          i        ,, and       a          i      +      1        &amp;gt;      a    i  , the coefficient             gcd      (              a                  i                    ,              a                  i          +          1                    )                      a                  i          +          1                    −              a                  i                     is   ≤  1. Thus we may bound the series above by      ∑          i      =      1              ∞            (          1              a                  i                      −          1              a                  i          +          1                      )    =      1          a              1              ≤  1.It follows that the finite sum is always strictly less than 1.

How to prove sum_(i=1)^(n-1) 1/(lcm(a_i, a_(i+1))<1 where a_i in NN and a_i<a_(i+1) ?

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