What have I done wrong in this summation formula?

Question

In Abel's summation formula we know that $$\sum_{1\leq n\leq x}^{} a(n)f(n)=A(x)f(x)-\int_{1}^{x}A(t)f'(t)dt$$
where $$A(x)=\sum_{1\leq n\leq x}^{}a(n)$$
Now let $a(n)=n , f(n)=\ln {n}$
We have $A(x)=1+2+...+x=\frac{x(x+1)}{2}$ and $\int_{1}^{x}A(t)f'(t)dt=\int_{1}^{x}\frac{t+1}{2}dt=x^2/4+x/2-3/4$

So, we should have $$\sum_{1\leq n\leq x}^{} n\ln {n}=\frac{x(x+1)}{2}\ln {x}-\frac{x^2}{4}-\frac{x}{2}+\frac{3}{4}$$
which is obviously false (take $x=4$ for example)

What have I done wrong?

Answer 1

$$ A(x)=1+2+...+x=\frac{x(x+1)}{2} $$ This is false when $x$ is not an integer: notice that the examples in the Wikipedia article require $A(x)$ to be a discontinuous function of the variable $x$ which is equal to the sum up to $\lfloor x \rfloor$, not the interpolation you have made.