, LectureO
Sums & Products
12 +
22 +... + n
=
1 .
2
..... n
= : = n !
Gauss Sum : 1 +... + n
==(
Geometric sum : 1-
1 - x
Perturbation method
S = 1 + x + x2 +... + xh
n +1
( .
S = x+ x2 +... + C
and consider the difference S-CS
AsymptoticBehaviors
,
limin 1
Variations
computeix" =
* 1 -
x
Thuric"1-n . x)2(n-1)
-1
and multiply by
+
(1 -
Sum of powers
(ent) (n 1) n
=
+ .
6
Approximation
= f(i) f(n)
&fi e
I f(1) = I+
wea
+
with I = ( f(x)d
Similar if f(x) is
weakly decreasing
I +
f(n) = f(i) f(t) +
, MCS
Chap 14
1 2
=
3
(14 1) .
+ +
-
1 xn+ 1
(14 2) 1+ x+ x2 +... x
-
. + =
1 -
x
Permutation Method
S = 1 + x + x2 +... + xh
1
xS = x + x + x3 +... + xn +
S = 1 + x+ x2 +... x
1
-
(S =
-
x -
x2 -
...
- xn +
1
xn
+
S -
xS =
1 -
solving s ,
closed form expression
1
S(1 x)
+
-
=
1 -
xn
1
xn
+
1
S
-
=
1 -
x
Closed form for Annuity value
Using previous equation ,
formula for V ,
value
of an
annuity that
pays m dollars
at the start each
of year for n
years
V =
m(
m(1 (1/(1 p))")
+
+
p
-
=
p
Sums & Products
12 +
22 +... + n
=
1 .
2
..... n
= : = n !
Gauss Sum : 1 +... + n
==(
Geometric sum : 1-
1 - x
Perturbation method
S = 1 + x + x2 +... + xh
n +1
( .
S = x+ x2 +... + C
and consider the difference S-CS
AsymptoticBehaviors
,
limin 1
Variations
computeix" =
* 1 -
x
Thuric"1-n . x)2(n-1)
-1
and multiply by
+
(1 -
Sum of powers
(ent) (n 1) n
=
+ .
6
Approximation
= f(i) f(n)
&fi e
I f(1) = I+
wea
+
with I = ( f(x)d
Similar if f(x) is
weakly decreasing
I +
f(n) = f(i) f(t) +
, MCS
Chap 14
1 2
=
3
(14 1) .
+ +
-
1 xn+ 1
(14 2) 1+ x+ x2 +... x
-
. + =
1 -
x
Permutation Method
S = 1 + x + x2 +... + xh
1
xS = x + x + x3 +... + xn +
S = 1 + x+ x2 +... x
1
-
(S =
-
x -
x2 -
...
- xn +
1
xn
+
S -
xS =
1 -
solving s ,
closed form expression
1
S(1 x)
+
-
=
1 -
xn
1
xn
+
1
S
-
=
1 -
x
Closed form for Annuity value
Using previous equation ,
formula for V ,
value
of an
annuity that
pays m dollars
at the start each
of year for n
years
V =
m(
m(1 (1/(1 p))")
+
+
p
-
=
p
