# Generalized Harmonic Summation

Evaluate

$\large{\sum_{r=1}^{\infty} \dfrac{H_{r} ^{(m)}}{r^m}} \quad ; \quad m \geq 2$

Notation: $$H_{r} ^{(m)}$$ denotes the Generalized Harmonic Number.

This is a part of the set Formidable Series and Integrals

Note by Ishan Singh
1 year, 10 months ago

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I have solved it by $$4$$ methods. One is Summation By Parts, other are the following :

First, expand the summation, using the definition of $$\displaystyle H_{r}^{(m)}$$ , to see that $$\displaystyle \sum_{r=1}^{n} \dfrac{H_{r}^{(m)}}{r^m} = \sum_{r=1}^{n} \dfrac{1}{r^{2m}} + \sum_{r < j} \sum_{j=2}^{n} \dfrac{1}{rj} = H_{n}^{(2m)} + \sum_{r < j} \sum_{j=2}^{n} \dfrac{1}{rj}$$

Also,

$$\displaystyle [H_{n}^{(m)}]^2 = H_{n}^{(2m)} + 2\sum_{r < j} \sum_{j=2}^{n} \dfrac{1}{rj}$$

Eliminating $$\displaystyle \sum_{r < j} \sum_{j=2}^{n} \dfrac{1}{rj}$$ from the above equations, we have,

$\displaystyle \sum_{r=1}^{n} \dfrac{H_{r}^{(m)}}{r^m} =\dfrac{1}{2} \left( [H_{n}^{(m)}]^2 + H_{n}^{(2m)} \right) \tag{*}$

Now take limit to infinity to get the desired result.

Also note that $$(*)$$ holds for $$m=1$$ as well (but the infinite sum diverges).

My other two methods use integral representations of $$\displaystyle H_{r}^{(m)}$$ and $$\dfrac{1}{r^m}$$. For instance, I have used one of them here.

- 1 year, 10 months ago

Yes even I thought about the same method! Shuffling indeed helps.

- 1 year, 10 months ago

$$\displaystyle S=\lim _{ n\rightarrow \infty }{ \sum _{ r=1 }^{ n }{ \frac { { H }_{ r }^{ \left( m \right) } }{ { r }^{ m } } } }$$

By Summation by parts, we get:

$$\displaystyle S=\lim _{ n\rightarrow \infty }{ { H }_{ n }^{ \left( m \right) }{ H }_{ n+1 }^{ \left( m \right) }-\sum _{ r=1 }^{ n }{ { H }_{ r }^{ \left( m \right) }\left( { H }_{ r+1 }^{ \left( m \right) }-{ H }_{ r }^{ \left( m \right) } \right) } }$$

Now we use $${ H }_{ r+1 }^{ \left( m \right) }={ H }_{ r }^{ \left( m \right) }+\frac { 1 }{ { \left( r+1 \right) }^{ m } }$$

$$\displaystyle S=\lim _{ n\rightarrow \infty }{ { H }_{ n }^{ \left( m \right) }{ H }_{ n+1 }^{ \left( m \right) }-\sum _{ r=1 }^{ n }{ \frac { { H }_{ r }^{ \left( m \right) } }{ { \left( r+1 \right) }^{ m } } } }$$

Again, we use $${ H }_{ r+1 }^{ \left( m \right) }={ H }_{ r }^{ \left( m \right) }+\frac { 1 }{ { \left( r+1 \right) }^{ m } }$$

$$\displaystyle S=\lim _{ n\rightarrow \infty }{ { H }_{ n }^{ \left( m \right) }{ H }_{ n+1 }^{ \left( m \right) }-S+1+\sum _{ r=1 }^{ n }{ \frac { 1 }{ { \left( r+1 \right) }^{ 2m } } } }$$

On simplifying, we get:

$\boxed{S=\frac { { \zeta }^{ 2 }\left( m \right) +\zeta \left( 2m \right) }{ 2 } }$

- 1 year, 10 months ago

What do you mean by 'summation by parts'?

- 1 year, 10 months ago

It is analogous to "Integration By Parts".

- 1 year, 10 months ago

Please correct me if I'm wrong.

- 1 year, 10 months ago

(+1) Correct!

- 1 year, 10 months ago

What was your method?

- 1 year, 10 months ago

Nice note!

- 1 year, 10 months ago

I now one value of this.

Which is the first one.

- 1 year, 10 months ago