We can find the number of terms needed to fall within that accuracy. Taking the absolute difference of the
th and
th partial sums, you have
or equivalently,
It's easy enough to check that
exceeds 10000 for all
, since
. This means the sum can be approximated to within
with just the first three terms.
You have
The actual value of the sum is
, which comes from the fact that
and plugging in
. The actual value is approximately
which verifies that the approximation is within 4 decimal places of the infinite series.