There are ways to approximate a sum through integration like the Euler–Maclaurin formula, which requires the function f(x) to be continuous, but there are several ways to extend the formula to functions with singularities:

1. Buchheit and Keßler - Singular Euler–Maclaurin expansion \ The singular Euler–Maclaurin expansion
2. Waterman, Yos, and Abodeely - Numerical Integration of Non-analytic Functions.

For me, it's not clear if the mentioned rules are acceptable and if not, what happens with the approximation after the application of the formula to the following functions, if one (or more) of the points in the sum are points of discontinuity for a given function (assuming f is continuous):

1. $\Gamma(x)$
2. $\ln\Gamma(x)$
3. $\ln(f(x))$
4. $\frac{1}{f(x)^2}$

The other question is: is it possible to determine if there is a point of singularity in a sum, for example, $\sum_{a}^{b}\ln(f(x))$, without computing the sum componentwise?

There are ways to approximate a sum through integration like the Euler–Maclaurin formula, which requires the function $f(x)$ to be continuous, but there are several ways to extend the formula to functions with singularities:

1. Buchheit and Keßler - Singular Euler–Maclaurin expansion (See also their post at the Wolfram Blog: The singular Euler–Maclaurin expansion)
2. Waterman, Yos, and Abodeely - Numerical Integration of Non-analytic Functions.

For me, it's not clear if the mentioned rules are acceptable and if not, what happens with the approximation after the application of the formula to the following functions, if one (or more) of the points in the sum are points of discontinuity for a given function (assuming $f$ is continuous):

1. $\Gamma(x)$
2. $\ln\Gamma(x)$
3. $\ln(f(x))$
4. $\dfrac{1}{f(x)^2}$

The other question is: is it possible to determine if there is a point of singularity in a sum, for example, $\sum_{a}^{b}\ln(f(x))$, without computing the sum componentwise?

There are ways to approximate a sum through integration like the Euler-Maclaurin formula, which requires the function f(x) to be continuous, but there are several ways to extend the formula to functions with singularities:

1. https://arxiv.org/abs/2003.12422 \ https://blog.wolfram.com/2021/06/30/the-singular-euler-maclaurin-expansion-a-new-twist-to-a-centuries-old-problem/
2. https://onlinelibrary.wiley.com/doi/epdf/10.1002/sapm196443145?saml_referrer

For me, it's not clear if the mentioned rules are acceptable and if not, what happens with the approximation after the application of the formula to the following functions, if one (or more) of the points in the sum are points of discontinuity for a given function (assuming f is continuous):

1. $\Gamma(x)$
2. $\ln\Gamma(x)$
3. $\ln(f(x))$
4. $\frac{1}{f(x)^2}$

The other question is: is it possible to determine if there is a point of singularity in a sum, for example, $\sum_{a}^{b}\ln(f(x))$, without computing the sum componentwise?

There are ways to approximate a sum through integration like the Euler–Maclaurin formula, which requires the function f(x) to be continuous, but there are several ways to extend the formula to functions with singularities:

1. Buchheit and Keßler - Singular Euler–Maclaurin expansion \ The singular Euler–Maclaurin expansion
2. Waterman, Yos, and Abodeely - Numerical Integration of Non-analytic Functions.

For me, it's not clear if the mentioned rules are acceptable and if not, what happens with the approximation after the application of the formula to the following functions, if one (or more) of the points in the sum are points of discontinuity for a given function (assuming f is continuous):

1. $\Gamma(x)$
2. $\ln\Gamma(x)$
3. $\ln(f(x))$
4. $\frac{1}{f(x)^2}$

The other question is: is it possible to determine if there is a point of singularity in a sum, for example, $\sum_{a}^{b}\ln(f(x))$, without computing the sum componentwise?