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Accelerated convergence is the term used to describe the replacement of a sequence by another that converges more quickly to the same limit value .
There are a number of different methods of accelerating convergence to choose from depending on the properties of the original sequence. Typical applications are iterative calculations, the evaluation of series and integration ( Romberg method ).
definition
One episode
T
=
{
t
n
}
n
∈
N
0
{\ displaystyle T = \ {t_ {n} \} _ {n \ in \ mathbb {N} _ {0}}}
with the limit converges faster than any other sequence
s
{\ displaystyle s}
S.
=
{
s
n
}
n
∈
N
0
{\ displaystyle S = \ {s_ {n} \} _ {n \ in \ mathbb {N} _ {0}}}
with the same limit, if the limit
lim
n
→
∞
‖
t
n
-
s
‖
‖
s
n
-
s
‖
{\ displaystyle \ lim _ {n \ to \ infty} {\ frac {\ | t_ {n} -s \ |} {\ | s_ {n} -s \ |}}}
exists and is zero. Obtained from a convergent sequence by a sequence transformation of the shape
T
{\ displaystyle T}
S.
{\ displaystyle S}
T
=
F.
(
S.
)
{\ displaystyle T = F (S)}
,
so one speaks of convergence acceleration.
example
The sequence converges with the order of convergence as against . The asymptotic development applies
a
n
=
∑
k
=
1
n
1
k
2
{\ displaystyle a_ {n} = \ sum _ {k = 1} ^ {n} {\ frac {1} {k ^ {2}}}}
1
n
{\ displaystyle {\ frac {1} {n}}}
π
2
6th
{\ displaystyle {\ frac {\ pi ^ {2}} {6}}}
∑
k
=
1
n
1
k
2
=
π
2
6th
-
1
n
+
1
2
n
2
-
1
6th
n
3
+
1
30th
n
5
-
1
42
n
7th
+
1
30th
n
9
-
5
66
n
11
+
O
(
1
n
13
)
,
n
→
∞
{\ displaystyle \ sum _ {k = 1} ^ {n} {\ frac {1} {k ^ {2}}} = {\ frac {\ pi ^ {2}} {6}} - {\ frac { 1} {n}} + {\ frac {1} {2n ^ {2}}} - {\ frac {1} {6n ^ {3}}} + {\ frac {1} {30n ^ {5}} } - {\ frac {1} {42n ^ {7}}} + {\ frac {1} {30n ^ {9}}} - {\ frac {5} {66n ^ {11}}} + {\ mathcal {O}} \! \ Left ({\ frac {1} {n ^ {13}}} \ right), \ quad n \ to \ infty}
This asymptotic series generates the Bernoulli numbers .
The terms in the sum of the series under consideration can pass
for k> 1
1
k
(
k
+
1
)
<
1
k
2
<
1
(
k
+
1
)
(
k
-
1
)
{\ displaystyle {\ frac {1} {k (k + 1)}} <{\ frac {1} {k ^ {2}}} <{\ frac {1} {(k + 1) (k-1 )}}}
be estimated. The rows for the estimates on the left and right are telescope rows ,
3
2
-
1
n
+
1
≤
1
+
∑
k
=
2
n
1
k
2
≤
7th
4th
-
n
+
1
2
n
(
n
+
1
)
{\ displaystyle {\ frac {3} {2}} - {\ frac {1} {n + 1}} \ leq 1+ \ sum _ {k = 2} ^ {n} {\ frac {1} {k ^ {2}}} \ leq {\ frac {7} {4}} - {\ frac {n + {\ frac {1} {2}}} {n (n + 1)}}}
.
The difference between the last two terms is
∑
k
=
2
n
1
k
2
-
1
-
∑
k
=
2
n
1
k
2
=
∑
k
=
2
n
1
k
2
(
k
2
-
1
)
{\ displaystyle \ sum _ {k = 2} ^ {n} {\ frac {1} {k ^ {2} -1}} - \ sum _ {k = 2} ^ {n} {\ frac {1} {k ^ {2}}} = \ sum _ {k = 2} ^ {n} {\ frac {1} {k ^ {2} (k ^ {2} -1)}}}
Thus also applies
π
2
6th
=
7th
4th
-
∑
k
=
2
∞
1
k
2
(
k
2
-
1
)
{\ displaystyle {\ frac {\ pi ^ {2}} {6}} = {\ frac {7} {4}} - \ sum _ {k = 2} ^ {\ infty} {\ frac {1} { k ^ {2} (k ^ {2} -1)}}}
.
The n -th partial sum of the series occurring in it converges with the order of convergence as , so much faster.
1
n
3
{\ displaystyle {\ frac {1} {n ^ {3}}}}
This process can be continued at will, so the difference between the last row and the telescope row can be observed.
∑
k
=
2
∞
1
(
k
-
1
)
k
(
k
+
1
)
(
k
+
2
)
{\ displaystyle \ sum _ {k = 2} ^ {\ infty} {\ frac {1} {(k-1) k (k + 1) (k + 2)}}}
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