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Methods that sum the same sequences (possibly to different limits); in other words, equivalent summation methods are summation methods having the same [[Summability field|summability field]]. Sometimes methods that have the same summability field and that are compatible summation methods (cf. [[Compatibility of summation methods|Compatibility of summation methods]]) are called equivalent. Examples of equivalent and compatible methods are the [[Cesàro summation methods|Cesàro summation methods]]  $  ( C, k) $
and the Riesz summation methods (cf. [[Riesz summation method|Riesz summation method]])  $  ( R, n, k) $(
for the same  $  k \geq  0 $), 
the Cesàro summation methods  $  ( C, k) $
and the [[Hölder summation methods|Hölder summation methods]]  $  ( H, k) $
(for the same integer  $  k \geq  0 $). 
There are equivalent summation methods that are not compatible.

Sometimes one considers not the complete summability fields, but subsets of them belonging to some set  $  U $. 
If these subsets coincide for two summation methods, then it is said that the methods are equivalent on  $  U $. 
Summation methods for real sequences are called completely equivalent if the equality of their summability fields remains valid upon the inclusion of sequences summable to  $  + \infty $
and  $  - \infty $. 
Equivalence of summation methods for special forms of summability (absolute, strong, etc.) is defined similarly.

[[Matrix summation method|Matrix summation method]]s, defined by transformations of sequences to sequences using matrices  $  \| a _ {nk} \| $
and  $  \| b _ {nk} \| $, 
are called absolutely equivalent on a set  $  U $
of sequences  $  \{ s _ {k} \} $
if  $  \tau _ {n}  ^ {( A)} - \tau _ {n}  ^ {( B)} \rightarrow 0 $, 
$  n \rightarrow \infty $, 
for any  $  \{ s _ {k} \} \subset  U $, 
where

$$ 
\tau _ {n}  ^ {( A)}  = \ 
\sum _ {k = 1 } ^  \infty  
a _ {nk} s _ {k} ,\ \ 
\tau _ {n}  ^ {( B)}  = \ 
\sum _ {k = 1 } ^  \infty  
b _ {nk} s _ {k} ,
$$

and the series in the expressions for  $  \tau _ {n}  ^ {( A)} $
and  $  \tau _ {n}  ^ {( B)} $
converge for all  $  n $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.G. Cooke,   "Infinite matrices and sequence spaces" , Macmillan  (1950)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.F. Kangro,   "Theory of summability of sequences and series"  ''J. Soviet Math.'' , '''5''' :  1  (1976)  pp. 1–45  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''12'''  (1974)  pp. 5–70</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Zeller,   W. Beekmann,   "Theorie der Limitierungsverfahren" , Springer  (1970)</TD></TR></table>