Unlike a decreasing geometric series, the sum of the harmonic series 1,1/2,1/3,1/4,1/5,......

Unlike a decreasing geometric series, the sum of the harmonic series 1,1/2,1/3,1/4,1/5,... diverges; that is,

It turns out that, for large n, the sum of the first n terms of this series can be well approximated as

where ln is natural logarithm (log base e = 2.718...) and γ is a particular constant 0.57721.... Show that

(Hint: To show an upper bound, decrease each denominator to the next power of two. For a lower bound, increase each denominator to the next power of 2.)