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Let's say I wanted to find $$\sum_{2 \leq n \leq x} f(n)$$

This is self evidently equal to $$f(2) + \sum_{2 < n \leq x} f(t)$$ which (by Euler's summation formula) is equal to equation(1):

$$f(2) + \int_2^x{f(t)dt} + \int_2^x(t - \lfloor t \rfloor)f'(t)dt +f(x)(\lfloor x \rfloor - x) -f(y)(\lfloor 2 \rfloor -2)$$

The last term is self 0 and can be discarded. Taking the integral $\int_2^x(\lfloor t \rfloor - t)f'(t)dt$ we have:

$$\int_2^x(t - \lfloor t \rfloor)f'(t)dt = \int_2^xtf'(t)dt - \int_2^x\lfloor t \rfloor f'(t)dt$$

Taking $\int_2^xtf'(t)dt$ we have through integration by parts, where $u=t$ and $f'(t)=dv$

$$\int_2^xtf'(t)dt = tf(t)\Big|_2^x - \int_2^xf(t)dt$$

And substituting these results into equation 1 gives: $$f(2) + \int_2^x{f(t)dt} + tf(t)\Big|_2^x - \int_2^xf(t)dt +\int_2^x \lfloor t \rfloor f'(t)dt +f(x)(\lfloor x \rfloor - x) -f(y)(\lfloor 2 \rfloor -2) =\\ f(2)+ tf(t)\Big|_2^x +\int_2^x\lfloor t \rfloor f'(t)dt + f(x)(\lfloor x \rfloor - x)$$

And since $$\int_2^x\lfloor t \rfloor f'(t)dt = 2\int_2^3f'(t) + 3\int_3^4f'(t)\ldots + (x-1)\int_{x-1}^xf'(t) = 2(f(3)-(f2))+3(f(4)-f(3))\ldots+(x-1)(f(x)-f(x-1)) = -2f(2)-f(3)-f(4)\ldots+(x-1)f(x)$$ we have:

$$\sum_{2 \leq n \leq x} f(n) = f(2)+ xf(x) - 2f(2) -2f(2)-f(3)-f(4)\ldots+(x-1)f(x) + f(x)(\lfloor x \rfloor - x) = f(2) - f(3) - f(4) +(2x-1)f(x) + f(x)(\lfloor x \rfloor - x)$$

so then: $$\sum_{2 \leq n \leq x} f(n) = f(2) - \sum_{3 \leq m \leq (x-1)}f(m) + (2x-1)f(x) + f(x)(\lfloor x \rfloor - x)$$

I can't quite put my finger on why, but this result seems\ldots suspect. Is my result correct? And if not, why not, and where did I make an error?

1 Answers 1

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Let's check with the simplest possible case: suppose that $f(n) = 1$, and take $x = 10$.

Then the LHS is $$ \sum_{n = 2}^{10} 1 = 9.$$

The RHS is $$ 1 - \sum_{n = 3}^9 1 + (20 - 1)\cdot 1 - 1\cdot(10 - 10) = 1 - 7 + 19 - 0 = 13.$$

So you are correct, there is an error somewhere. I didn't track the error down, but one way to do so would be to insert the $f(n) = 1$ function into your attempted proof and see where an inequality pops up.