Let H_1, H_2, H_3, ... be harmonic numbers. For n in Z+, the sum sum_{j=1}^n…
2004
Let H_1, H_2, H_3, ... be harmonic numbers. For n in Z+, the sum sum_{j=1}^n H_j can be expressed as:
- A.
nH_{n+1} - (n + 1)
- B.
(n + 1)H_n - n
- C.
nH_n - n
- D.
(n + 1)H_{n+1} - (n + 1)
Show answer & explanation
Correct answer: B
We use H_j = 1 + 1/2 + ... + 1/j. Then sum_{j=1}^n H_j = sum_{j=1}^n sum_{k=1}^j 1/k. Interchange the order of summation. For a fixed k, the term 1/k appears for j = k, k+1, ..., n, so it appears n-k+1 times. Hence sum_{j=1}^n H_j = sum_{k=1}^n (n-k+1)/k = (n+1)sum_{k=1}^n 1/k - sum_{k=1}^n 1 = (n + 1)H_n - n.