Unlike a decreasing geometric series, the sum of the harmonic series 1; 1=2; 1=3; 1=4; 1=5; : : : diverges; that is, 1x i=1 1 i = 1: it turns out that, for large n, the sum of the rst n terms of this series can be well approximated as xn i=1 1 i ln n + ; where ln is natural logarithm (log base e = 2: 718 : : : ) and is a particular constant 0: 57721 : : : . show that xn i=1 1 i = (log n): (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.)