Can we prove that e − τ ≤ e − τ / t t − 1 for t ∈ ( 0 , 1 ) and 0 &lt; τ &lt; t ?

Question

Can we prove that      e          −      τ        ≤      e          −      τ              /            t            t          −      1      for   t  ∈  (  0  ,  1  ) and   0  &amp;lt;  τ  &amp;lt;  t ?

Isaias Archer · Accepted Answer

Let   x  =            τ      t        . Then   0  &amp;lt;  x  &amp;lt;  1. The inequality takes the form      e          −      x      t        ≤      e          −      x            t          −      1        t      e          x      (      1      −      t      )        ≤  1We have  t      e          x      (      1      −      t      )        ≤  t      e          1      −      t        t      e          x      (      1      −      t      )        ≤  t      e          1      −      t      Let y=1−t. It suffices to show that  t      e          1      −      t        =  (  1  −  y  )      e    y    ≤  1  ,    0  ≤  y  &amp;lt;  1We have      e    y    =      ∑          n      =      0        ∞                      y        n                    n        !              ≤      ∑          n      =      0        ∞        y    n    =  (  1  −  y      )          −      1      So the result follows.

Can we prove that e^-\tau<= e^(tau/t)t^-1 for t in (0,1) and 0<tau<t?

Answered question

Answer & Explanation

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