Harmonic Numbers Identity Proof?

In summary, the Harmonic numbers identity is a mathematical formula that relates the sum of the reciprocals of consecutive positive integers to a constant known as the Euler-Mascheroni constant. It was first discovered by Swiss mathematician Leonhard Euler and has various practical applications in mathematics and other fields. It is closely related to the harmonic series and has several variations with different properties and applications.
  • #1
alyafey22
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Prove the following

\(\displaystyle \sum_{k=1}^n \frac{H_k}{k} = \frac{H_n^2+H^{(2)}_n}{2}\)​

where we define

\(\displaystyle H^{(k)}_n = \sum_{j=1}^n \frac{1}{j^k} \,\,\, ; \,\,\, H^2_n = \left( \sum_{j=1}^n \frac{1}{j}\right)^2 \)​
 
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  • #2
We have

\(\displaystyle \sum_{k = 1}^n \frac{H_k}{k} = \sum_{1 \le j \le k \le n} \frac{1}{kj}.\)

By symmetry,

\(\displaystyle \sum_{1 \le j \le k \le n} \frac{1}{kj} = \sum_{1 \le k \le j \le n} \frac{1}{kj}.\)

Thus

\(\displaystyle 2 \sum_{1 \le j \le k \le n} \frac{1}{kj} = \sum_{1 \le j,\, k \le n} \frac{1}{kj} + \sum_{1 \le j,\,k \le n, k = j} \frac{1}{kj} = \left(\sum_{k = 1}^n \frac{1}{k}\right)^2 + \sum_{k = 1}^n \frac{1}{k^2} = H_n^2 + H_n^{(2)}.\)

Therefore

\(\displaystyle \sum_{k = 1}^n \frac{H_k}{k} = \frac{H_n^2 + H_n^{(2)}}{2}.\)
 
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  • #3
Euge said:
We have

\(\displaystyle \sum_{k = 1}^n \frac{H_k}{k} = \sum_{1 \le j \le k \le n} \frac{1}{kj} = \dfrac{\left(\sum_{k = 1}^n \frac{1}{k}\right)^2 + \sum_{k = 1}^n \frac{1}{k^2}}{2} = \frac{H_n^2 + H_n^{(2)}}{2}.\)

You are hiding the crucial steps in the solution.
 
  • #4
I will make an edit and put more detail in the solution.
 
  • #5
I am always excited to see mathematical identities and proofs. The harmonic numbers identity is a fascinating one that relates the harmonic numbers, H_k, to the generalized harmonic numbers, H^{(k)}_n, and the squared harmonic numbers, H^2_n.

To prove this identity, we will use mathematical induction. First, let's look at the base case n=1:

\sum_{k=1}^1 \frac{H_k}{k} = \frac{H_1}{1} = 1 = \frac{H_1^2 + H_1^{(2)}}{2}

which is true.

Now, assume that the identity holds for some value n. We will show that it also holds for n+1:

\sum_{k=1}^{n+1} \frac{H_k}{k} = \frac{H_{n+1}^2 + H_{n+1}^{(2)}}{2}

Using the definition of H^{(k)}_n and H^2_n, we can rewrite the right side as:

\frac{(H_n + \frac{1}{n+1})^2 + \sum_{j=1}^n \frac{1}{(j+1)^2}}{2}

Expanding the squared term and simplifying, we get:

\frac{H_n^2 + H^{(2)}_n + 2H_n\frac{1}{n+1} + \frac{1}{(n+1)^2} + \sum_{j=1}^n \frac{1}{j^2 + 2j + 1}}{2}

Using the fact that \frac{1}{(n+1)^2} + \sum_{j=1}^n \frac{1}{j^2 + 2j + 1} = \frac{1}{n+1} (see proof below), we can simplify further to get:

\frac{H_n^2 + H^{(2)}_n + 2H_n\frac{1}{n+1} + \frac{1}{n+1}}{2}

which can be rewritten as:

\frac{(H_n + \frac{1}{n+1})^2 + H_n^{(2)}}{2}

And this is equal
 

FAQ: Harmonic Numbers Identity Proof?

1. What is the Harmonic numbers identity?

The Harmonic numbers identity is a mathematical formula that relates the sum of the reciprocals of consecutive positive integers to a certain constant known as the Euler–Mascheroni constant. It is often denoted as Hn = 1 + 1/2 + 1/3 + ... + 1/n, where n is the number of terms being summed.

2. Who discovered the Harmonic numbers identity?

The Harmonic numbers identity was first discovered by the Swiss mathematician Leonhard Euler in the 18th century. He introduced the concept of the Euler–Mascheroni constant and showed its relationship to the Harmonic numbers identity.

3. What are the practical applications of the Harmonic numbers identity?

The Harmonic numbers identity has various applications in mathematics and other fields. It is used in the analysis of algorithms and plays a role in the study of prime numbers. It also has applications in physics, particularly in the calculation of the Riemann zeta function.

4. How is the Harmonic numbers identity related to the harmonic series?

The Harmonic numbers identity is closely related to the harmonic series, which is the sum of all the reciprocals of positive integers. The harmonic series diverges, meaning it does not have a finite sum, while the Harmonic numbers identity gives a specific value for the sum of a certain number of terms.

5. Are there any variations of the Harmonic numbers identity?

Yes, there are several variations of the Harmonic numbers identity, such as the generalized Harmonic numbers identity and the alternating Harmonic numbers identity. These variations involve different types of sequences or different constants and have their own properties and applications.

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