To solve the expression \( \frac{5}{13} \pm \frac{5}{17} \pm \frac{21}{41} + \frac{8}{13} \pm \frac{20}{41} \), we first need to clarify if the positive and negative signs are to be applied individually, as they are written. If they are interchangeable, we can't evaluate the expression without knowing which sign to choose for each term.

However, we can break down the expression into separate parts and compute them independently using one combination of signs to demonstrate the process:

1. Combine the fractions:
\( \frac{5}{13} + \frac{8}{13} = \frac{5 + 8}{13} = \frac{13}{13} = 1 \)

2. Now, let's consider the other fractions:
\( \frac{5}{17}, \frac{21}{41}, \) and \( \frac{20}{41} \).

For simplicity, let’s choose all plus signs for now, so we evaluate:
\[
1 + \frac{5}{17} + \frac{21}{41} + \frac{20}{41}
\]

Combine \(\frac{21}{41}\) and \(\frac{20}{41}\):
\[
\frac{21 + 20}{41} = \frac{41}{41} = 1
\]

Thus, we have:
\[
1 + \frac{5}{17} + 1 = 2 + \frac{5}{17}
\]

To combine \(2\) and \(\frac{5}{17}\), convert \(2\) to a fraction:
\[
2 = \frac{34}{17}
\]

Now add:
\[
\frac{34}{17} + \frac{5}{17} = \frac{34 + 5}{17} = \frac{39}{17}
\]

So, when considering all plus signs, the result is:
\[
\frac{39}{17}
\]

**Final Result with all plus signs:** \(\frac{39}{17}\)

If you need a different combination of signs (some negatives), please specify, and I can calculate that too!