diff --git a/src/sage/rings/lazy_series.py b/src/sage/rings/lazy_series.py
index 51eeaf5d065..c9e079702bc 100644
--- a/src/sage/rings/lazy_series.py
+++ b/src/sage/rings/lazy_series.py
@@ -584,7 +584,8 @@ def map_coefficients(self, f):
 
     def truncate(self, d):
         r"""
-        Return this series with its terms of degree >= ``d`` truncated.
+        Return the series obtained by removing all terms of degree at least
+        ``d``.
 
         INPUT:
 
@@ -1989,7 +1990,7 @@ def sec(self):
 
     def arcsin(self):
         r"""
-        Return the arcsin of ``self``.
+        Return the arcsine of ``self``.
 
         EXAMPLES::
 
@@ -2021,7 +2022,7 @@ def f(n):
 
     def arccos(self):
         r"""
-        Return the arccos of ``self``.
+        Return the arccosine of ``self``.
 
         EXAMPLES::
 
@@ -2116,7 +2117,7 @@ def arccot(self):
 
     def sinh(self):
         r"""
-        Return the sinh of ``self``.
+        Return the hyperbolic sine of ``self``.
 
         EXAMPLES::
 
@@ -2144,7 +2145,7 @@ def sinh(self):
 
     def cosh(self):
         r"""
-        Return the cosh of ``self``.
+        Return the hyperbolic cosine of ``self``.
 
         EXAMPLES::
 
@@ -2171,7 +2172,7 @@ def cosh(self):
 
     def tanh(self):
         r"""
-        Return the tanh of ``self``.
+        Return the hyperbolic tangent of ``self``.
 
         EXAMPLES::
 
@@ -3388,9 +3389,9 @@ def __call__(self, g, *, check=True):
         Given two Laurent series `f` and `g` over the same base ring, the
         composition `(f \circ g)(z) = f(g(z))` is defined if and only if:
 
-        - `g = 0` and `val(f) >= 0`,
+        - `g = 0` and `\mathrm{val}(f) \geq 0`,
         - `g` is non-zero and `f` has only finitely many non-zero coefficients,
-        - `g` is non-zero and `val(g) > 0`.
+        - `g` is non-zero and `\mathrm{val}(g) > 0`.
 
         INPUT:
 
@@ -3574,7 +3575,7 @@ def __call__(self, g, *, check=True):
             y
 
         We look at cases where the composition does not exist.
-        `g = 0` and `val(f) < 0`::
+        `g = 0` and `\mathrm{val}(f) < 0`::
 
             sage: g = L(0)
             sage: f = z^-1 + z^-2
@@ -3585,7 +3586,7 @@ def __call__(self, g, *, check=True):
             ...
             ZeroDivisionError: the valuation of the series must be nonnegative
 
-        `g \neq 0` and `val(g) \leq 0` and `f` has infinitely many
+        `g \neq 0` and `\mathrm{val}(g) \leq 0` and `f` has infinitely many
         non-zero coefficients::
 
             sage: g = z^-1 + z^-2
@@ -3797,7 +3798,7 @@ def revert(self):
 
         The compositional inverse exists if and only if:
 
-        - `val(f) = 1`, or
+        - `\mathrm{val}(f) = 1`, or
 
         - `f = a + b z` with `a, b \neq 0`, or
 
@@ -3870,7 +3871,7 @@ def revert(self):
             ...
             ValueError: compositional inverse does not exist
 
-        `val(f) != 1` and `f(0) * f(1) = 0`::
+        `\mathrm{val}(f) != 1` and `f(0) * f(1) = 0`::
 
             sage: (z^2).revert()
             Traceback (most recent call last):
@@ -4287,7 +4288,7 @@ def exponential(self):
 
     def compute_coefficients(self, i):
         r"""
-        Computes all the coefficients of self up to i.
+        Computes all the coefficients of ``self`` up to ``i``.
 
         This method is deprecated, it has no effect anymore.
 
@@ -4343,12 +4344,12 @@ def __call__(self, *g, check=True):
         Given a Taylor series `f` of arity `n` and a tuple of Taylor
         series `g = (g_1,\dots, g_n)` over the same base ring, the
         composition `f \circ g` is defined if and only if for each
-        `1\leq k\leq n`:
+        `1\leq i\leq n`:
 
         - `g_i` is zero, or
-        - setting all variables except the `i`th in `f` to zero
+        - setting all variables except the `i`-th in `f` to zero
           yields a polynomial, or
-        - `val(g_i) > 0`.
+        - `\mathrm{val}(g_i) > 0`.
 
         If `f` is a univariate 'exact' series, we can check whether
         `f` is a actually a polynomial.  However, if `f` is a
@@ -4621,7 +4622,7 @@ def revert(self):
 
         The compositional inverse exists if and only if:
 
-        - `val(f) = 1`, or
+        - `\mathrm{val}(f) = 1`, or
 
         - `f = a + b z` with `a, b \neq 0`
 
@@ -4681,7 +4682,7 @@ def revert(self):
             ...
             ValueError: compositional inverse does not exist
 
-        `val(f) != 1` and `f(0) * f(1) = 0`::
+        `\mathrm{val}(f) != 1` and `f(0) * f(1) = 0`::
 
             sage: (z^2).revert()
             Traceback (most recent call last):
@@ -5284,13 +5285,13 @@ def __call__(self, *args, check=True):
         Given a lazy symmetric function `f` of arity `n` and a tuple
         of lazy symmetric functions `g = (g_1,\dots, g_n)` over the
         same base ring, the composition (or plethysm) `(f \circ g)`
-        is defined if and only if for each `1\leq k\leq n`:
+        is defined if and only if for each `1\leq i\leq n`:
 
         - `g_i = 0`, or
-        - setting all alphabets except the `i`th in `f` to zero
+        - setting all alphabets except the `i`-th in `f` to zero
           yields a symmetric function with only finitely many
           non-zero coefficients, or
-        - `val(g) > 0`.
+        - `\mathrm{val}(g) > 0`.
 
         If `f` is a univariate 'exact' lazy symmetric function, we
         can check whether `f` has only finitely many non-zero
@@ -5489,7 +5490,7 @@ def revert(self):
 
         The compositional inverse exists if and only if:
 
-        - `val(f) = 1`, or
+        - `\mathrm{val}(f) = 1`, or
 
         - `f = a + b p_1` with `a, b \neq 0`.