While studying mechanics, I had an idea for a novel way of finding the sum of all squares from 1 to n. I thought that if in mechanics, we had an object that travelled at a velocity of 1 m/s at t=1, 5 m/s at t=2 and so on we'd complete a table just like the below and expect to find a formula for the velocity of the object at any given time by using calculus.
However, when I've tried this method, I have failed, and I cannot see why other than the typical cop-out of "you cannot use calculus on a problem about integers", which I've largely dismissed because as I've mentioned, if you saw these numbers as some data about a body's velocity at a given time, you'd expect it to work regardless of the fact that all of your data points were integers.
Regardless, here is what I have tried so far. Firstly I have completed the table below (again, it's the same trick as in mechanics, if f(n) was an object's velocity, we'd all know how I've found g(n), which would be the acceleration).
Once finishing the table, I have once again continued as we could in mechanics.
i(n) is clearly the constant function i(n)=2 $\forall$ n.
Integrating this gives h(n)=2n+c for some constant c. Substituting in n=2 gives c=1, so h(n)=2n+1.
Continuing on the same path for g(n), we get g(n)=$n^2 + n + 0.25$.
Then for f(n), we get f(n)=$\frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{4} + c$ which upon substitution, reveals that we must have made an error. n=1 would give c=$ - \frac{1}{12}$ while n=2 gives c=$ - \frac{1}{6}$ which is clearly inconsistent.
So, where is the error? Why can't this method be used in this case, when we would've expected it to work if we encountered the same numbers in a mechanics problem?