Download - General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Singular Points v2

8/9/2019 General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Si…

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2/20

x = 1

1x+1

= 1−x+x2−x3+x4− ... 12 = 1−1+1−1+1−...

n 1−1 + 1−1 + 1− ...

12

1 + 2 + 3 + 4 + 5 + ... = −112

+∞i=1 ai

sn = n

i=1 ai


3/20

x = −1 11−x =1 + x + x2 + x3 + x4 + ...

1 − 1 + 1 − 1 + ... + (−1)n

−1

+ ... =

1

2 .

x = 2

1 + 2 + 4 + 8 + ... + 2n−1 + ... = −1. x = 1 1+1+1+1+1+... = 10 x = −2

1 − 2 + 4 − 8 + ... + (−2)n−1 + ... = 13

.

x = a |a| > 1

1 + a + a2 + a3 + ... + an−1 + ... = 1

1 − a.


4/20

x = 1 11+x = 1 − 2x + 3x2 − 4x3 + 5x4 − ...

1 − 2 + 3 − 4 + ... + (−n)n

−1

+ ... =

1

4 .

x = −1 1+2+3+4+5+... = 102 −1 +∞

ζ (s) = 1−s + 2−s + 3−s + 4−s + 5−s + ... (∗)

Re(s) > 1 2−s

(1

−21−s)

·(1−s + 2−s + 3−s + 4−s + 5−s + ...) = 1−s

−2−s + 3−s

−4−s + 5−s + ... (

∗∗)

s = −1 −3 · ( 1 + 2 + 3 +4 + 5 + ...) = 1 − 2 + 3 − 4 + ... −3 · (1 + 2 + 3 + 4 + 5 + ...) = 1

4

1 + 2 + 3 + 4 + ... + n + ... = −1

12 .

s = 0

1 + 1 + 1 + 1 + ... + n0 + ... = −1

2 .

2 + 3 + 4 + 5 + ... + (n + 1) + ... = −7

12 .

0 + 1 + 2 + 3 + ... + (n − 1) + ... = 512

.

1−s ln1 + 2−s ln2 + 3−s ln3 + 4−s ln 4 +5−s l n 5 + ... = −ζ (s) 1−s ln 1 − 2−s l n 2 + 3−s ln 3 − 4−s l n 4 + 5−s ln 5 − ... = (21−s −1)ζ (s) − 21−s ln 2 · ζ (s) s = 0

ln 1 + ln 2 + ln 3 + ln 4 + ... + ln(n) + ... = 1

2 ln(2π),

ln 1 − ln2 + ln 3 − ln4 + ... + (−1)n−1 ln(n) + ... = −12

ln(1

2π).

s = 1 + x + x2 + x3 + x4 + ... = 1 + x · s


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s = 11−x 1 + x + x2 + x3 + x4 + ... = 11−x

x = 1 x = eiθ 0 < θ


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s = 1

1−s + 2−s + 3−s + 4−s + ... + n−s + ... = ζ (s)(Re(s) < 1).

limn→+∞ s1+s2+...+snn s1 = 1, s2 = 1 + (−1) = 0, s3 = 1 + (−1) + 1 = 1, s4 =1 + (−1)+1+(−1) = 0,...

s11 =

11 = 1,

s1+s22 =

1+02 =

12 ,

s1+s2+s33 =

1+0+13 =

23 ,

s1+s2+s3+s44 =

1+0+1+04 =

12

, s1+s2+s3+s4+s55

= 1+0+1+0+15

= 35

,... 12

1 + (−1) + 0 + 1 + (−1) + 0 + ... = 23

,

1 + (−1) + 0 + 1 + (−1) + 0 + ... = 13

.

x = 1

1+x1+x+x2

+∞n=0(−1)n

= +∞n=0(−1)

n

· 1 =+∞n=0(−1)n·

+∞0 e

−ttndt

n! =

+∞n=0

+∞0 e

−t (−t)nn!

dt = +∞0 e

−t+∞n=0

(−t)nn!

dt = +∞0 e

−2tdt =12

1 − 1! + 2! − 3! + 4! − ... + (−1)n−1(n − 1)! + ... = 0, 596347...,

1 + 1! + 2! + 3! + 4! + ... + (n − 1)! + ... = 0, 697175...,

x

k=1 f (x) =

c + +∞0 f (t)dt + 12f (x) ++∞k=1 B2k(2k)!f (2k−1)(x)

c = − 12f (0) −+∞

k=1B2k(2k)!

f (2k−1)(0) f (t) 1, t , t + 1 t−1 c B2 = 16 x = −1 − ln(1+x) 1 + 12 +

13 +

14 +

15 + ... = − ln 0

1 + 1

2 +

1

3 +

1

4 +

1

5 + ... +

1

n + ... = γ = 0.57721566...


7/20

s =+∞

k=1 ak sn =

nk=1 ak

+∞

k=1

ak = limn→+∞

n

k=1

ak.

sn

s1+s2+...+snn

limn→+∞

sn = limn→+∞

s1 + s2 + ... + snn

.

sn sn ±∞

sn

sn =s(n) {1, 2, 3, ...n, } {ω(1), ω(2), ω(3), ...ω(n)}

ω (7)

1 + 1 + 1 + 1 + ...n0 + ... = −12

.

sn = s(n) = n

limDn→+∞ sω(n) = limDn→+∞ ω(n) = lim

Dn→+∞

ω(1)+ω(2)+ω(3)+...+ω(n)n

=s =

−12

ω(k) = − kn + 1

limDn→+∞1n

nk=1 ω(k) =

−12

limDn→+∞1n

nk=1 s(k) = lim

Dn→+∞

nk=1 s(− kn+1)· 1n =

0−1 s(k)dk ∈

R

D

limn→+∞

sn =

0−1

s(n)dn.


8/20

n −∞ limDn→−∞ sn =limDn→+∞ s−n =

0

−1 s(−n)dn D

limn→−∞

sn =

10

s(n)dn.

s(n) [1, +∞) ∞ [−1, 0) 0 s

[−1, 0) limn→−∞ sn s(n) [0, 1) I (x) = −1/x I : [1, +∞) → [−1, 0) I : [−∞, −1) → [0, 1)

(6)

1 + 2 + 3 + 4 + ... + n + ... = s.

sn = n(n+1)

2

s = Dlimn→+∞

sn = 0−1

s(n)dn = 0−1

n(n + 1)2

dn = −1/12.

f ∞ ∞ E R = {z ∈ C ||z | > R} ∞ BR = BR(∞) = E R

{∞} r > 0 a B(a; r) = {z |0


9/20

∞ f ◦ J 0 I (z ) = −J (z ) = − 1

z f

∞

f ◦ I E R f B(a; 1) P ∞(0, 0, 1) P 0(0, 0, −1) E R B(a; 1)

∞

∞

lα =

{r

·eiα

|r

∈ R

}, α

∈ [0, 2π),

∞

∞

∞ C f (z ) z 0 ω ∈ C

{∞} z n → z 0 f (z n) → ω n → +∞ ∞


10/20

+∞k=−∞ ck(z ) · 1zk

+∞k=−∞ ck(z ) · (z − a)k

a ∈ C

f (z ) ∞ f (z ) ∞ lα = {r · eiα|r ∈ R} limDz→∞(α) f (z ) f (z ) ∞ lα

D

limz→∞(α)+

f (z ) =D

limz→+eiα∞

f (z ) =D

limr→+∞

f (r · eiα),

D

limz→∞(α)−

f (z ) =D

limz→−eiα∞

f (z ) =D

limr→−∞

f (r · eiα),D

limz→∞(α)

f (z ) =D

limr→∞(0)

f (r · eiα),D

limz→∞(α)

f (z ) = 1

2(

D

limz→∞(α)+

f (z ) +D

limz→∞(α)−

f (z )).

f (z ) ∞

limDz→∞

(α) f (z ) α

∈ [0, 2π) limDz

→∞f (z )

α = 0 l0 = R

D

limx→∞(0)+

f (x) =D

limx→+∞

f (x),

D

limx→∞(0)−

f (x) =D

limx→−∞

f (x).


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f (z ) a ∈ C f (z ) a lα,a = {a+r ·eiα|r ∈ R} limD

z→a(α)f (z )

f (z ) a lα,a

D

limz→a(α)+

f (z ) =D

limz→a+eiα0

f (z ) =D

limr→0+

f (a + r · eiα),

D

limz→a(α)−

f (z ) =D

limz→a−eiα0

f (z ) =D

limr→0−

f (a + r · eiα),D

limz→a(α)

f (z ) =D

limr→0(0)

f (a + r · eiα),

Dlimz→a(α)

f (z ) = 12

( Dlimz→a(α)+

f (z ) + Dlimz→a(α)−

f (z )).

f (z ) a

limDz→a(α) f (z ) α ∈ [0, 2π) limDz→a f (z ) α = 0 l0 = R

D

limx→a(0)+

f (x) =D

limx→a+

f (x),

D

limx→a(0)

−f (x) =

D

limx→a

−f (x).

f (z ) a ∈ C {∞}

D

limz→a

f (z ) =D

limz→a(α)±

f (z ) = limz→a

f (z ), α ∈ [0, 2π).

f (z ) = z n (n ∈ N ) z → ∞(0)

D

limz→+∞

z n = 0

−1z ndz,

D

limz→−∞

z n =

10

z ndz.

f (z ) = z −n (n ∈ N ) z → 0(0)

D

limz→0+

z −n = −1−∞

z −n−2dz,


12/20

D

limz→0−

z −n = +∞1

z −n−2dz.

f (z )

a ∈ C {∞}

a c0(z ) = f (z ) a

d(c0(z )) = c0(z ) f (z ) g(z ) a ∈ C {∞} c ∈ C

Dlim

z→a(α)±(f (z ) + g(z )) =

Dlim

z→a(α)±f (z ) +

Dlim

z→a(α)±g(z ),

D

limz→a(α)±

(c · f (z )) = c ·D

limz→a(α)±

f (z ).

ex sin(x) cos(x) ∞ limx→+∞ ex = +∞ limx→+∞ sin(x) limx→+∞ cos(x) +∞

e

x

+ 0

sin(x) + 0

cos(x) + 0

lim

D

x→+∞ ex

= lim

D

x→+∞ sin(x) =limDx→+∞ cos(x) = 0

∞k=0 anx

n 0

limDx→+∞ ax = 0(a = 0)

ln(x) ∞ limx→+∞ ln(x) =+∞ +∞ ln(x) + 0 limDx→+∞ ln(x) = 0 lim

Dx→+∞

n√

xm

=0, (m, n) = 1

limz→

+∞

H z = +∞

f (z ) = H z

∞ c0(z ) = H z ∞

d(H z)dz = H z

16

(π2 − 6H (2)z )dz = H z,

H n =n

k=11k

= ψ(0)(n + 1) + γ

H (s)n =

nk=1

1ks

= ζ (s) − ζ (s, n + 1) ζ (s, z) ≡+∞k=0 1(k+z)s


13/20

ψ(0)(z + 1) + C = H z,

C = H z − ψ(0)(z + 1),C = γ.

limDz→+∞ H z = γ.

f (z ) a ∈ C {∞(0)} f (z ) =F 1(z )+F 2(z ) a F 1(z ) F 2(z ) c0

D

limz→a(α)±

f (z ) =D

limz→a(α)±

F 1(z ) + c0.

a ∈ C limDz→a(α)± f (z ) = limDz→a(α)±(F 1(z )+F 2(z )) = limDz→a(α)±(−1

k=−n ck·(z −a)k+∞

k=0 ck·(z −a)k) = limDz→a(α)±

−1k=−n ck·(z −a)k+limz→a

∞k=0 ck·(z −a)k = limDz→a(α)± F 1(z )+

c0 = limDz→a(α)±

nk=1 c−k · (z −a)−k + c0 = limDr→0(0)±

nk=1 c−k ·(a + r−ke−iαk−a)−k +

c0 = n

k=1 c−k · e−iαk limDr→0(0)± r−k + c0 = n

k=1 c−k · e−iαk(∓ ∓∞∓1 r

−k−2dr) + c0 =nk=1 c−k · e−iαk (±1)

k

k+1 + c0 =

−1k=−n ck · eiαk (±1)

k

2(−k+1) + c0. a = ∞ limDz→±∞ f (z ) =

nk=1 ck · eiαk (±1)

k

2(k+1) + c0

a ∈ C lim

Dz→a(α) f (z ) =

12(lim

Dz→∞(α)+ f (z )+lim

Dz→∞(α)− f (z )) =

12(−1k=−n ck ·eiαk (+1)

k

−k+1 + c0 +−1k=−n ck · eiαk (−1)

k

−k+1 + c0) =−1

k=−n ck · eiαk 1+(−1)k

2(−k+1) + c0. a = ∞ limDz→∞(α) f (z ) =

nk=1 ck · eiαk 1+(−1)

k

2(k+1) + c0

f (z ) z = a ∈ C {∞} c0 f (z ) a

D

limz→a(α)

f (z ) = c0.

ζ (z ) z = 1 z = 1 1

z−1 + γ − γ 1(z − 1) + 12γ 2(z − 1)2 − 16γ 3(z − 1)3 + 124γ 4(z − 1)4 + O((z − 1)5).

limDz→1(0) ζ (z ) = limDz→1(0)(F 1(z ) + c0) = lim

Dz→1(0)(

1z−1 + γ ) = 0 + γ = γ.

ψ(0)(z) = Γ(z)Γ(z) Γ(x) =

+∞0 tx−1e−tdt


14/20

f (x) a ∈ C {∞} f (x) =F 1(x) + F 2(x) a c0

D

limz→a

f (z ) = c0.

α ∈ [0, 2π) a ∈ C limDz

→a f (z ) =

12π

· 2π

0 limDz→a(α) f (z )dα =

12π

· 2π

0 −1k=−n ck121+(−1)k

−k+1

eiαkdα + c0 =12π

·−1k=−n ck 12 1+(−1)k−k+1 2π0 eiαkdα + c0 = 0 + c0 = c0. a = ∞

ζ (z ) z = 1 z = 1 1

z−1 + γ − γ 1(z − 1) + 12γ 2(z − 1)2 − 16γ 3(z − 1)3 + 124γ 4(z − 1)4 + O((z − 1)5).

limDz→1 ζ (z ) = c0 = γ.

1 − 1 + 1 − 1 + ... + (−1)n−1 + ... = 12

.

sn = 12 · ((−1)n+1 + 1) = 12 · (−1)n+1 + 12 s = 12 .

1 + 2 + 4 + 8 + ... + 2n−1 + ... = −1.

sn = 2n − 1 s = −1.


15/20

1 − 2 + 4 − 8 + ... + (−2)n

−1

+ ... =

1

3 .

sn = −13 · ((−2)n − 1) = −13 · (−2)n + 13 s = 13 .

1 + a + a2 + a3 + ... + an−1 + ... = 1

1 − a.

sn = 1a−1 · (an − 1) = 1a−1 · an − 1a−1 s = 11−a .

1 − 2 + 3 − 4 + ... + (−n)n−1 + ... = 14

.

sn = −14 · ((−1)n(2n + 1) − 1) = −14 · (−1)n(2n + 1) + 14 s = 14 .

1 + 2 + 3 + 4 + ... + n + ... = −1

12 .

sn = n(n+1)

2 s = limDn→+∞

n(n+1)2 =

0−1

n(n+1)2 dn = − 112 .

1 + 1 + 1 + 1 + ... + n0 + ... = −1

2 .

sn = n s = limDn→

+∞

n = 0−1 ndn = −12

.

2 + 3 + 4 + 5 + ... + (n + 1) + ... = −7

12 .

sn = n(n+3)

2 s = limDn→+∞

n(n+3)2

= 0−1

n(n+3)2

dn = − 712

.


16/20

0 + 1 + 2 + 3 + ... + (n − 1) + ... = 5

12 .

sn = n(n−1)

2 s = limDn→+∞

n(n−1)2 =

0−1

n(n−1)2 dn =

512 .

ln 1 + ln 2 + ln 3 + ln 4 + ... + ln(n) + ... = 1

2 ln(2π).

sn = ln((1)n) (1)n = Γ(1+n)

Γ(1)

s(n) = sn n = ∞ ln(2−

arg(z+1)+π2π

cscarg(z+1)+π

2π (π(z +1))+arg(z+1)+π2π (iπz + iπ2 +O((1z )7))+arg(z)+π2π (2iπz +

iπ + O((1z

)7)) + ((− ln(1z

) − 1)z + 12

(ln(2π) − ln(1z

)) + 112z

− 1360z3

+ 11260z5

+ O((1z

)6))) s = 12 ln(2π).

ln 1 − ln2 + ln 3 − ln4 + ... + (−1)n−1 ln(n) + ... = −12

ln(1

2π).

sn = −12(−1)n ln(2)+(−1)n ln(Γ(n+12 )) − (−1)n ln(Γ(n+22 )) − 12 ln(2π) +ln(2) s = −12 ln(2π) + ln(2) = − 12 ln( 12π).

cos θ + cos 2θ + cos 3θ + cos 4θ + ... + cos(nθ) + ... = −12

.

sn = 12 cot( θ2) sin(nθ) + 12 cos(nθ) − 12 s = −12 .

sin θ + sin 2θ + sin 3θ + sin 4θ + ... + sin(nθ) + ... = 1

2 cot

θ

2.

(x)n = Γ(x+n)

Γ(x) = x(x + 1) · · · (x + n − 1)


17/20

sn = −12 cot(θ2)cos(nθ) + 12 sin(nθ) + 12 cot( θ2) s = 12 cot θ2 .

cos θ − cos2θ + cos 3θ − cos4θ + ... + (−1)n−1 cos(nθ) + ... = 12

.

sn = 12(cos((θ + π)(n + 1)) + tan

θ2 · sin((θ + π)(n + 1)) + 1), s = 12 .

sin θ − sin2θ + sin 3θ − sin4θ + ... + (−1)n−1 sin(nθ) + ... = 12

tan θ

2.

sn = 12(− tan θ2 cos((θ + π)n) − sin((θ + π)n) + tan θ2) s = 12 tan θ2 .

12k − 22k + 32k − 42k + ... + (−1)n−1n2k + ... = 0(k = 1, 2, 3,...).

sn = 22k(−1)n+1ζ (−2k, n+12 ) + 22k(−1)nζ (−2k, n+22 ) − 22k+1ζ (−2k) +

ζ (−2k) s = (1 − 22k+2)ζ (−2k) = 0.

12k+1−22k+1+32k+1−42k+1+ ... +(−1)n−1n2k+1+ ... = 22k+2 − 12k + 2

B2k+2(k = 0, 1, 2,...).

sn = 22k+1(−1)n+1ζ (−2k − 1, n+1

2 ) + 22k+1(−1)nζ (−2k − 1, n+2

2 ) −

22k+2ζ (−2k − 1) + ζ (−2k − 1) s = (1 − 22k+2)ζ (−2k − 1) = 22k+2−12k+2 B2k+2.

1k + 2k + 3k + 4k + ... + nk + ... = − Bk+1k + 1

(k = 1, 2, 3,...).

sn = 1k+1

km=0

k+1m

Bmn

k+1−m B1 =

12 .

s = limDn→+∞( 1k+1

km=0

k+1m

Bmn

k+1−m) = 1k+1

km=0

k+1m

Bm lim

Dn→+∞ n

k+1−m =1

k+1

km=0

k+1m

Bm

0−1 n

k+1−mdn = − 1k+1

km=0

k+1m

Bm

(−1)k+2−mk+2−m = − 1k+1 · Bk+1.


18/20

B1

= 12

B1

= −

1

2

Bk+1 = 1 −k

m=0

k + 1

m

Bm

k + 2 − m,

Bk+1 = −k

m=0

k + 1

m

Bm

k + 2 − m.

1−s + 2−s + 3−s + 4−s + ... + n−s + ... = ζ (s)(Re(s) < 1).

sn = H (s)n = ζ (s) − ζ (s, n + 1) S = ζ (s)

1 + (−1) + 0 + 1 + (−1) + 0 + ... = 23

.

sn = 23 + n3 − n3 s = 23 .

1 + (−1) + 0 + 1 + (−1) + 0 + ... = 13

.

sn = −−23 + n3 + −13 + n3 s = 13 .

x ∈ R\Z x = −12 + x +∞

k=1sin(2kπx)

k

π

1 − 1! + 2! − 3! + 4! − ... + (−1)n−1(n − 1)! + ... = 0, 596347...

sn = −e((−1)nE n+1(1)Γ(n+1)+Ei(−1)) s = −eEi(−1) = 0, 596347...

E n(x) = +∞1

e−xtdttn

Ei(x) = − +∞−x

e−tdtt


19/20

1 + 1! + 2! + 3! + 4! + ... + (n−

1)! + ... = 0, 697175...

sn = (−1)nn!!(−n − 1)+!(−2) + 1 s =!(−2) + 1 = 0, 697175...

1 + 1

2 +

1

3 +

1

4 +

1

5 + ... +

1

n + ... = γ = 0.57721566...

sn = ln(n) + γ + εn εn ∼ 12n s = γ.

sn = H n = ψ(0)(n + 1) + γ s = γ.

+∞0

sin xdxD = 1,

+∞0

ln x sin xdxD = −γ,

+∞0

√ x tanh(

√ x)dxD = −3

4ζ (3).

+∞0

sin xdxD = 1.

!n = Γ(n+1,−1)e

Γ(a, x) = +∞x

ta−1e−tdt


20/20

+∞0 sin xdx

D = − cos x|+∞0 = −(limDx→+∞(cos x + 0) − cos 0) = −(0 −1) = 1.

+∞0

ln x sin xdxD = −γ.

+∞0

ln x sin xdxD = (Ci(x)−ln x cos x)|+∞0 = limDx→+∞(Ci(x)−ln x cos x)−limDx→0(Ci(x) − ln x cos x) = 0 − γ = −γ. Ci(x)− ln x cos x ∞ cos x(ln 1

x+O((1

x)7))+cos x(−( 1

x)2+ 6

x4− 120

x6 +O(( 1

x)7))+sin x( 1

x− 2

x3 + 24

x5 +O(( 1

x)7))+

O(( 1x

)9)−

iπ1

2 − arg(x)

π + 0.

Ci(x)− ln x cos x 0 γ + 14x

2(2ln x − 1) + 196x4(1 − 4 ln x) + x6(6lnx−1)

4320 + O(x7).

+∞0

√ x tanh(

√ x)dxD = −3

4ζ (3).

+∞0

√ xh(

√ x)dxD = (−2√ xLi2(−e−2

√ x)−Li3(−e−2

√ x)+2x

32

3 +2x ln(e−2

√ x+

1))|+

∞0 = limD

x→+∞(−2√ xLi2(−e−2√ x

) − Li3(−e−2√ x

) + 2x

32

3 + 2x ln(e−2√ x

+1) +0) −3ζ (3)4

= 0 − 3ζ (3)4

= −3ζ (3)4

.

Ci(x) = − +∞x

cos tdtt

Lin(z) ≡+∞

k=1zk

kn