Is there a method to find the value of the $n$-th decimal place of $\pi$ which is more efficient than having to compute all decimal places before as well?



Compute an arbitrary decimal place of $\pi$
Yes, there are such algorithms.
-- See e.g. Xavier Gourdon: [Computation of the $n$-th decimal digit of $\pi$ with low memory](http://numbers.computation.free.fr/Constants/Algorithms/nthdecimaldigit.pdf).
There also is the Bailey-Borwein-Plouffe formula already mentioned by Geoff Robinson in a comment -- see <https://www.math.hmc.edu/funfacts/ffiles/20010.5.shtml>.


