1

My textbook has the steps to prove it, but I can't comprehend the steps that the textbook are showing. Can someone explain the math or logic used going from steps red to yellow and finally green?

enter image description here

Belphegor
  • 1,268
  • 6
  • 27
  • 51

1 Answers1

2

To get from red to yellow, note that $$(k+1)H_k = (k+1)\sum_{j=1}^k \frac{1}{j} = -(k+1)\frac{1}{k+1} + (k+1)\sum_{j=1}^{k+1}\frac{1}{j} = (k+1)\left(H_{k+1}-\frac{1}{k+1}\right).$$ To get from yellow to green, just expand: \begin{align} (k+1) \left(H_{k+1}-\frac{1}{k+1}\right) - k + H_{k+1} &= (k+1)H_{k+1} - 1 - k + H_{k+1} \\ &= (k+2)H_{k+1} - 1 - k. \end{align}

rogerl
  • 22,399
  • Wherr did the negative come from for (k+1)? – Belphegor Jul 04 '14 at 18:23
  • To get from the first expression in the first equation to the second, we added $\frac{1}{k+1}$ to the sum (check the indices on the summation). The $-(k+1)\frac{1}{k+1}$ is to correct for that. – rogerl Jul 04 '14 at 18:25