For the regular Sturm-Liouville problem with equation
y
+
(
λ
−
q
(
x
)
)
y
=
0
y + (\lambda - q(x))y = 0
on
0
≤
x
≤
π
0 \le x \le \pi
, there are well-known asymptotic expansions for the eigenvalues and eigenfunctions. We show that these asymptotic expansions can be replaced by convergent series for sufficiently large eigenvalues. Convergence is uniform on the interval
0
≤
x
≤
π
0 \le x \le \pi
and uniform with respect to the eigenvalues, in the sense that a single majorant bounds all series. The basic idea is to replace the asymptotic results, which use an expansion of powers of
n
−
1
o
r
(
n
+
1
/
2
)
−
1
{n^{ - 1}}or{(n + 1/2)^{ - 1}}
for integers
n
n
, by a series in powers of
μ
−
1
{\mu ^{ - 1}}
, where
μ
2
{\mu ^2}
is an eigenvalue for the corresponding constant coefficient Sturm-Liouville problem with equation
y
+
λ
y
=
0
y + \lambda y = 0
.