The Euler-Maclaurin summation formula is as follows: $$\sum_{i=m}^{n} f(i) = \int_{m}^{n} f(x) dx + \frac{f(n)+f(m)}{2} + \sum_{k=1}^{\lfloor p/2 \rfloor} \frac{B_{2k}}{(2k)!}\big{(}f^{(2k-1)}(n)-f^{(2k-1)}(m)\big{)} + R_{p} .$$

Usually, it yields an approximation, because the third term (the sum involving the Bernoulli numbers) and the remainder term cannot be evaluated exactly.

If the higher derivatives eventually become zero at the start and end points, the formula becomes exact. An example of an application of this fact is Faulhaber's formula.

However, Faulhaber's formula pertains to finite sums. I wonder whether there are infinite series for which all terms of the formula above can be evaluated. In particular, I would like to know whether some systematic study on the circumstances under which such an evaluation is possible and exact has been published.

I've asked a similar question a while ago on MSE.

Added note. To clarify, by 'exact' I mean 'has a closed form expression'

The Euler–Maclaurin summation formula is as follows: $$\sum_{i=m}^{n} f(i) = \int_{m}^{n} f(x) dx + \frac{f(n)+f(m)}{2} + \sum_{k=1}^{\lfloor p/2 \rfloor} \frac{B_{2k}}{(2k)!}\big{(}f^{(2k-1)}(n)-f^{(2k-1)}(m)\big{)} + R_{p} .$$

Usually, it yields an approximation, because the third term (the sum involving the Bernoulli numbers) and the remainder term cannot be evaluated exactly.

If the higher derivatives eventually become zero at the start and end points, the formula becomes exact. An example of an application of this fact is Faulhaber's formula.

However, Faulhaber's formula pertains to finite sums. I wonder whether there are infinite series for which all terms of the formula above can be evaluated. In particular, I would like to know whether some systematic study on the circumstances under which such an evaluation is possible and exact has been published.

I've asked a similar question In what case(s) does the Euler-Maclaurin summation method yield the exact evaluation? a while ago on MSE.

Added note. To clarify, by ‘exact’ I mean ‘has a closed form expression’.

The Euler-Maclaurin summation formula is as follows: $$\sum_{i=m}^{n} f(i) = \int_{m}^{n} f(x) dx + \frac{f(n)+f(m)}{2} + \sum_{k=1}^{\lfloor p/2 \rfloor} \frac{B_{2k}}{(2k)!}\big{(}f^{(2k-1)}(n)-f^{(2k-1)}(m)\big{)} + R_{p} .$$

Usually, it yields an approximation, because the third term (the sum involving the Bernoulli numbers) and the remainder term cannot be evaluated exactly.

If the higher derivatives eventually become zero at the start and end points, the formula becomes exact. An example of an application of this fact is Faulhaber's formula.

However, Faulhaber's formula pertains to finite sums. I wonder whether there are infinite series for which all terms of the formula above can be evaluated. In particular, I would like to know whether some systematic study on the circumstances under which such an evaluation is possible and exact has been published.

I've asked a similar question a while ago on MSE.

The Euler-Maclaurin summation formula is as follows: $$\sum_{i=m}^{n} f(i) = \int_{m}^{n} f(x) dx + \frac{f(n)+f(m)}{2} + \sum_{k=1}^{\lfloor p/2 \rfloor} \frac{B_{2k}}{(2k)!}\big{(}f^{(2k-1)}(n)-f^{(2k-1)}(m)\big{)} + R_{p} .$$

Usually, it yields an approximation, because the third term (the sum involving the Bernoulli numbers) and the remainder term cannot be evaluated exactly.

If the higher derivatives eventually become zero at the start and end points, the formula becomes exact. An example of an application of this fact is Faulhaber's formula.

However, Faulhaber's formula pertains to finite sums. I wonder whether there are infinite series for which all terms of the formula above can be evaluated. In particular, I would like to know whether some systematic study on the circumstances under which such an evaluation is possible and exact has been published.

I've asked a similar question a while ago on MSE.

Added note. To clarify, by 'exact' I mean 'has a closed form expression'