How to define the $r$th harmonic number if $r$ is any real number? What would be the value of 

$$\sum_{k=1}^{\sqrt{\frac{5}{2}}}\frac{1}{k}$$

Is it $H_{\sqrt{\frac{5}{2}}}$?
Using different definitions of harmonic numbers this question can be computed, but can I use the usual definition of harmonic numbers for any real numbers?
In other words is 
$$\sum_{k=1}^{n}\frac{1}{k}$$
A useful definition for any real $n$?
When I want to compute the value of $H_{\sqrt{\frac{5}{2}}}$ at Desmos I just define $\sum_{k=1}^{n}\frac{1}{k}$ and I don't put $\sqrt{\frac{5}{2}}$ directly at the upper limit, but I define $n=\sqrt{\frac{5}{2}}$ and the result is exactly what it should be, it seems using a simple substitution will give us the right answer, but generally are we allowed to use the regular definition of harmonic numbers when our $n$ is not necessarily a natural number?
 A: Indefinite Sum  concept is the answer to your question.
In fact,  if a function $f(x)$ is the (forward) difference of a function $F(x)$
$$
f(x) = \Delta _x F(x) = F(x + 1) - F(x)
$$
then we say that $F(x)$ is the "antidifference" (or "indefinite sum") of $f(x)$
$$
F(x) = \Delta _x ^{\left( { - 1} \right)} f(x) = \sum\nolimits_x {f(x)} 
$$
If $ f(x)$ and $F(x)$ are defined over a real, or complex, domain for $x$ then we will have for example
$$
\sum\limits_{k = 0}^n {f(x + k)}  = \sum\limits_{k = 0}^n {\left( {F(x + 1) - F(x)} \right)}  = F(x + n + 1) - F(x)
$$
We write the above  with a different symbol for the sum as
$$
\sum\nolimits_{k = 0}^{\,n} {f(x + k)}  = \sum\limits_{k = 0}^{n - 1} {f(x + k)}  = F(x + n) - F(x)
$$
which can also be written as
$$
\sum\nolimits_{k = x}^{\,x + n} {f(k)}  = F(x + n) - F(x)
$$
The extension to 
$$
\sum\nolimits_{k = a}^{\,b} {f(k)}  = F(b) - F(a)
$$
for any real (or complex) $a,b$ inside the domain of definition of $f, F$ is quite natural.
Coming to the harmonic numbers,  it is well known that the functional equation of the digamma function is
$$
{1 \over x} = \Delta _x \,\psi (x)
$$
and it is therefore "natural" to define
$$ \bbox[lightyellow] {  
H_r  = \sum\limits_{k = 1}^r {{1 \over k}}  = \sum\nolimits_{k = 1}^{\,1 + r} {{1 \over k}}  = \psi (1 + r) - \psi (1) = \psi (1 + r) + \gamma 
 } \tag{1}$$
which substantiate M. Janisch's comment
Answering to W. Hintze's comment, please consider how the "indefinite sum" parallels the "indefinite integral - antiderivative" concept.
Similarly to
$$
f(x) = {d \over {dx}}F(x)\quad  \Leftrightarrow \quad F(x) = \int {f(x)dx}  + c
$$
we have that
$$
f(x) = \Delta \,F(x)\quad  \Leftrightarrow \quad F(x) = \sum\nolimits_x {f(x)}  + \pi \left( x \right)
$$
where now the family of  antidifference functions differ by any function $\pi(x)$ , and not by a constant, which is periodic
with  period (or one of the periods) equal to $1$, as you rightly noticed.
So by "natural extension" I meant to say:
 - an extension from integers to reals (and complex) field under the "indefinite sum" concept,
which provides a function $\mathbb C \to \mathbb C$ which fully interpolates $f(n)$;
 - in the antidifference family to select the "simpliest / smoothiest" function, same as 
the Gamma function is selected among the functions satisfying $F(z+1)=zF(z)$ as the
only one which is logarithmically convex,
or which has the simplest Weierstrass representation, etc.
From the comments I could catch some skepticism as if my answer could just be an "extravagant" personal
idea of mine.
 That's absolutely not so, the definition in (1) is actually standardly accepted: refer to 
Wolfram Function Site, 
and in particular to this section or to the
Wikipedia article.
I am just trying to enlight how we can assign a meaning to sums with bounds which are not integral and thus saying that 
$$ \bbox[lightyellow] {  
H_{\,1 + \sqrt {5/2} }  = \sum\limits_{k = 1}^{\,1 + \sqrt {5/2} } {{1 \over k}}  = \sum\nolimits_{k = 1}^{\,\,2 + \sqrt {5/2} } {{1 \over k}}  = \psi (\,\,2 + \sqrt {5/2} ) - \psi (1) = 1.7068 \ldots 
 } \tag{2}$$
A: 
The  $n$-th harmonic  number  $H_n=\sum_{k=1}^n\frac{1}{k}$ admits  a   nice representation  as  integral  of  a finite  geometric series  which can  be  generalised  in a rather natural way. We have  the following  representation  for  $n$ a positive integer:
  \begin{align*}
H_n=\sum_{k=1}^n\frac{1}{k}=\int_0^1\frac{1-t^n}{1-t}dt\tag{1}
\end{align*}

The  identity  (1) is valid since we have
\begin{align*}
\int_0^1\frac{1-t^n}{1-t}dt&=\int_0^1\left(1+t+\cdots+t^{n-1}\right)\,dt\tag{2}\\
&=\left(t+\frac{1}{2}t^2+\cdots+\frac{1}{n}t^n\right)\bigg|_0^1\tag{3}\\
&=\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)-\left(0\right)\tag{4}\\
&=H_n
\end{align*}
Comment:


*

*In (2) we apply  the  finite geometric series formula.

*In (3) we  do  the integration.

*In (4) we   evaluate the   expression  at the upper and  lower  limit.

The representation (1) indicates  a  generalisation
\begin{align*}
H_\color{blue}{r}=\int_0^1\frac{1-t^\color{blue}{r}}{1-t}dt\tag{2}
\end{align*}
  for   real  values $r$ in fact also for complex values.

Note:


*

*The formula (2) can be found in the wiki page generalized harmonic numbers.

*We have an interesting relationship with the Digamma function $\psi(r)$ which has an integral representation
\begin{align*}
\psi(r+1)=\int_0^1\frac{1-t^r}{1-t}\,dt-\gamma
\end{align*}
which is strongly related to (2). Here  $\gamma$ is the Euler-Mascheroni constant.

*If we stick at the sigma notation, we often find for real upper limit $r$ the meaning
\begin{align*}
\sum_{k=1}^r a_k=\sum_{k=1}^{\lfloor r\rfloor}a_k
\end{align*}
where $\lfloor .\rfloor$ is the floor function.
A: Your question is "are we allowed to use the regular definition of harmonic numbers when our $n$ is not necessarily a natural number?" The answer is no, since the "regular definition" of a sum is $a_1 + a_2 + ... + a_n$. Here $n$ is the number of summands which can't be a non natural number.
But you can very well give a meaning to the result of a sum. And this result may accept non integers as input.
Take the simpler example of a "sum" $a(\pi) = \text{"}\sum_{k=1}^{\pi} k\text{"}$. This "sum" is not defined, as although the first three summands are 1, 2, and 3, the last 3.14...-th summand is not defined. 
Hence the definition starts by calculating the sum up to an integer $n$, and afterwards insert the non integer value for $n$. Here this gives the arithmetic sum $a(n)= \frac{1}{2}n(n+1)$, and hence $a(\pi)= \frac{1}{2}\pi(\pi+1)$.
Now for the harmonic number there is no such simple closed expression for general $n$. But here comes a relation which is valid for natural $n$:  $\sum_{k=1}^n \frac{1}{k} = \sum_{k=1}^n \int_0^1 x^{k-1}\,dx=\int_0^1 \sum_{k=1}^n x^{k-1}\,dx=\int_0^1 \frac{1-x^n}{1-x}\,dx$, and now the r.h.s. is valid for also for real $n$.
Your description of the problem gives a good illustration of this procedure. In making the usual definition of the harmonic number $\sum_{k=1}^n \frac{1}{k}$ in your CAS, the system understands that you mean the harmonic number function $H_n$ it has in its "belly", and this is valid for real numbers. If you then insert a value for $n$, be it integer or not, you get the result of this internal function.
