Euler-Maclaurin Formula

[kindred kindle]

So I'm trying to prove the Euler Maclaurin Formula. So far I have proven this

[kindred kindle]

this is the formula in its entirely

and trhis is how the wiki defines the remainder term

the only reason my sum is slightly different is because I'm using B_1 = -1/2 instead of +1/2

[kindred kindle]

so according to this $$R_p=\sum_{k=p+1}^\infty\frac{B_k}{k!}\left(f^{(k-1)}(n)-f^{(k-1)}(m)\right)$$

[dreamy sluice]

kheerii

[kindred kindle]

and I've made this into $$R_p=\int_m^n \sum_{k=p+1}^\infty\frac{B_k}{k!}f^{(k)}(t)\dd{t}$$

[dreamy sluice]

kheerii

[kindred kindle]

but I don't know what else to do