Sum of n "integers"? I was reading a paper on the Gamma Function and came across the problem posed with regard to interpolation theory. I have no idea about interpolation theory, and a simple google search translated interpolation to estimation.
My question is with regard to the equation $a_n=\frac{(n)(n+1)}{2}$
We've all learnt in school how exactly we formed this equation. My question is, is n confined only to integer values? I'm having trouble understanding the following statement from Pg.3 in the above mentioned link:
"The formula extends the scope of the original problem to values of the variable other than those for which it was originally defined and solves the problem of interpolating between the known elementary values"
My reasoning is as follows:
We obviously know that the sum of n integers satisfies the formula. But what does it mean for n to equal a non integral value, let's say $4.5$. It obviously doesn't mean $1+2+3+4+(4.5)$ because 4.5 isn't even an integer. So, does this mean that the formula doesn't return a meaningful value for some n, not an integer?
If not so, on what basis are we saying that we're estimating the sum for the $4.5^{th}$ integer?
 A: It is very common for a concept in mathematics to have a larger scope than expected. Consider, for instance, the sine function. The definition of sine we are taught in elementary geometry as the ratio of the opposite side to the hypotenuse in a right triangle only applies when dealing with acute angles. Later, we discover that sine has many other properties, most of which go far beyond computing the angles and sides of triangles. To do this, however, we shift our perspective. $\sin(100^{\text{o}})$ can no longer mean the ratio of the opposite side to the hypotenuse; instead, it must refer to something different.
In a similar vein, the formula for the sum of first $n$ positive integers is
$$
S_n=\frac{n(n+1)}{2} \, ;
$$
but notice that it is completely meaningful to consider the function $f: n \mapsto n(n+1)/2$ even when $n$ is not a positive integer. So even though $f(4.5)$ does not refer to the sum of the first $4.5$ integers (whatever that would mean), it is still interesting to study the properties of the function $f$. It is almost certainly the case that $f$ will not have all the amazing properties that sine does, but that not does not mean our task in generalising the sum of the first $n$ positive integers was unfruitful.
The gamma function is a better example of such a generalisation, since it is widely documented how useful it was to generalise the factorial beyond integer values. The Riemann zeta function can be analytically continued to the complex plane, and the gamma function provides an explicit form for its value. Given the significance of the Riemann zeta function in number theory, don't you think this is amazing? These developments would not have been possible had it not been for our capacity to broaden our minds and think beyond the initial definition of a function.
