Kwanini 1+2+3+4+5+6...∞ ni sawa na -1/12 ??

"An equation for me has no meaning, unless it expresses a thought of God."
--Srinivasa Ramanujan

Good day good citizens🙂
Leo ni siku nzuri ya hesabu na leo nataka tuongelee infinite series yaani sum ya sequence ya namba zisizo na mwisho..

Post yangu iliyopita niliongelea maana ya series na aina za series
Sasa hivi nataka kuonesha kuwa series ya mjumuliso wa namba zote sio divergent bali ni convergent (refer to last post)
Yani kama S=1+2+3+4+5....∞ tukiendelea kujumlisha hivi mpaka milele tutapata jibu la -1/12

Proof:
ili kuprove hii kitu inabidi tuanze kujua series moja inaitwa grandi's series
Ambayo ipo hivi
X=1-1+1-1+1-1+1-1.....∞
Ukiangalia hiyo sequence kadri unapoendelea kujumlisha moja na kutoa moja popote utakaposimama jibu linakuja either 0 au 1
Hivyo kwakuwa sequence inaenda mpaka infinity na hatujui infinity itasimamia wapi basi jawabu la hii series inaonekana kuwa ni kati ya 0 na 1 hivyo tunasema jibu ni 0.5 au 1/2
Hivyo hii series ina converge to 1/2
X=1-1+1-1+1-1...∞ = 1/2
X=1/2

Pia tuangalie series nyingine
Y=1-2+3-4+5-6+7-8+9....∞
Hii series mara mbili ni sawa na 2Y
2Y = Y + Y
Hivyo tuzijumlishe hizi series kwa zenyewe

1-2+3-4+5-6+7-8+9....
+ 1-2+3-4+5-6+7-8.....
=1-1+1-1+1-1+1-1....
View attachment 1335424
Kwahiyo
2Y=1-1+1-1+1-1....
Lakini kumbuka X = 1-1+1-1+1-1+1...
2Y = X
2Y = 1/2
Y = 1/4

Sasa turudi Kwenye series yetu tuiite S
S= 1+2+3+4+5+6.....+∞
Kumbuka Y = 1-2+3-4+5-6+7...∞

Tutafute S - Y = ?
S - Y =
1+2+3+4+5+6+7+8+9+10..
- [1-2+3-4+5-6+7-8+9-10...]
=4+8+12+16+20....∞

Ukiangalia hizo namba ni multiples za 4
Kwahyo 4 tunaweza kuitoa nje ikawa

=4(1+2+3+4+5...∞)

Lakini 1+2+3+4...∞ = S
= 4S
Hivyo
S - Y = 4S

Kumbuka Y = 1/4
Hivyo
S - 1/4 = 4S
-1/4 = 3S
3S = -1/4
Therefore
S = - 1/12

Hivyo 1+2+3+4+5....∞ = -1/12

All credit kwa mwanamahesabu aliyejifunza mwenyewe bila kufundisha na kugundua vingi Kwenye hesabu Srinivasa Ramanujan
View attachment 1335428

"An equation for me has no meaning, unless it expresses a thought of God."
--Srinivasa Ramanujan

Good day good citizens🙂
Leo ni siku nzuri ya hesabu na leo nataka tuongelee infinite series yaani sum ya sequence ya namba zisizo na mwisho

Post yangu iliyopita niliongelea maana ya series na aina za series
Sasa hivi nataka kuonesha kuwa series ya mjumuliso wa namba zote sio divergent bali ni convergent (refer to last post)
Yani S=1+2+3+4+5....tukiendelea kujumlisha hivi mpaka milele tutapata jibu la -1/12

Proof:
ili kuprove hii kitu inabidi tuanze kujua series moja inaitwa grandi's series
Ambayo ipo hivi
X=1-1+1-1+1-1+1-1.....∞
Ukiangalia hiyo sequence kadri unapoendelea kujumlisha moja na kutoa moja popote utakaposimama jibu linakuja either 0 au 1
Hivyo kwakuwa sequence inaenda mpaka infinity na hatujui infinity itasimamia wapi basi jawabu la hii series inaonekana kuwa ni kati ya 0 na 1 hivyo tunasema jibu ni 0.5 au 1/2
Hivyo hii series ina converge to 1/2
X=1-1+1-1+1-1...∞ = 1/2
X=1/2

Pia tuangalie series nyingine
Y=1-2+3-4+5-6+7-8+9....∞
Hii series mara mbili ni sawa na 2Y
2Y = Y + Y
Hivyo tuzijumlishe hizi series kwa zenyewe

1-2+3-4+5-6+7-8+9....
+ 1-2+3-4+5-6+7-8.....
=1-1+1-1+1-1+1-1....

Kwahiyo
2Y=1-1+1-1+1-1....
Lakini kumbuka X = 1-1+1-1+1-1+1...
So
2Y = X
2Y = 1/2
Y = 1/4

Sasa turudi Kwenye series yetu tuiite S
S= 1+2+3+4+5+6.....+∞
Kumbuka Y = 1-2+3-4+5-6+7...∞

Tutafute S - Y = ?
S - Y =
1+2+3+4+5+6+7+8+9+10..
- [1-2+3-4+5-6+7-8+9-10...]
=4+8+12+16+20....∞

Ukiangalia hizo namba ni multiples za 4
Kwahyo 4 tunaweza kuitoa nje ikawa

=4(1+2+3+4+5...∞)

Lakini 1+2+3+4...∞ = S
= 4S
Hivyo
S - Y = 4S

Kumbuka Y = 1/4
Hivyo
S - 1/4 = 4S
-1/4 = 3S
3S = -1/4
Therefore
S = - 1/12

Hivyo 1+2+3+4+5....∞ = -1/12

Pioneered by Srinivasa Ramanujan mwanamahesabu wakihindi aliyetokea familia masikini bila kwenda shule na kujifundisha hesabu mwenyewe mpaka kugundua vitu vikubwa Kwenye pure mathematics.

Mtaalamu wa hesabu, fundi, role model wangu.
Srinivasa Ramanujan.

#ApumzikeKwaAmani

Sasa inamsadia nini binadamu kwenye maisha yake ya kawaida ili mkono uende kinywani?

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Beira Boy said:

Sasa inamsadia nini binadamu kwenye maisha yake ya kawaida ili mkono uende kinywani?

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Ukijua unalipwa...kuna swali ukijibu unapewa dola milioni moja

Nikiambiwa nichague Kati ya haya mahesabu na kambare nachagua kambare!!

Maana sielewi
Makambare tujuane twendeni tukaidhibu sakafu kwa kugalagala😜😜

Huo ni uongo mkubwa sana, ati;

1+2+3+4+5-----∞=-1/12

The sums of positive integers from 1 to infinite equals to a negative irrational number 0.08333333333-----????

-0.083333333333333----.

Kawadanganye wasiojua hesabu.🤣🤣

Wabongo mkila mkishiba mnatupa tabu sana wengine

Mokaze, I just proved it sasa sijui unataka nini tena
Ungeonesha labda wapi Kwenye proof nimekosea au ungeleta jibu lako na njia yako ungekuwa umefanya la maana

Citizen B said:

I just proved it sasa sijui unataka nini tena
Ungeonesha labda wapi Kwenye proof nimekosea au ungeleta jibu lako na njia yako ungekuwa umefanya la maana

Click to expand...

Mkuu, japo siwezi kuproove,.mathematically, lakini nina assume hicho ulichoweka hapo siyo kweli kabisaaaaa, nina assume kwakuwa hata Newton ali assume "inverse square law" katika gravitational equation.

1+2+3+4+5+6------∞=∞ that is my proof.


Hiyo ni arithmetic progression;

Let s=(1+2--------∞), therefore, s=(∞------2+1),

2S=∞(∞+1), therefore S=[∞(∞+1)]/2


Fanya hiyo ∞ iwe ni namba, hapo utaona hiyo series ina diverge to infinity

Mnatuchosha tu hamna kipya mnachogundua ni mi namba tu sasa inawasaidia nini

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Uongo mtupu kila kitu mmeassume assume tu..Najua Maths na napenda Maths ila siwezi kuburuzwa na theories chache za uongo cha wataalam wa zamani wa hesabu.
Kwenye kuprove hiyo series yako umeanza na series nyingine(grandi's series X=1-1+1-1+1-1+1-1.....∞ = 1/2 )
je imetokea wapi na je ipo valid? anza kwanza kuprove hiyo grandi. yaani unasema jibu ni either 0 au 1 halafu unaamua kuweka nusu, how?????

Halafu tuletee na proofs za wanaopinga ili tulinganishe

Aiseee.....

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Islam005 said:

Uongo mtupu kila kitu mmeassume assume tu..Najua Maths na napenda Maths ila siwezi kuburuzwa na theories chache za uongo cha wataalam wa zamani wa hesabu.
Kwenye kuprove hiyo series yako umeanza na series nyingine je imetokea wapi na je ipo valid?
Halafu tuletee na proofs za wanaopinga hiyo kitu ili tulinganishe

Click to expand...

Huyo jamaa anadanganya, na sio hapo tu hata upande wa dini pia anapenda kudanganya.

Mimi namjua kiasi fulani huyo Citizen b.

Mokaze said:

Huyo jamaa anadanganya, na sio hapo tu hata upande wa dini pia anapenda kudanganya.

Mimi namjua kiasi fulani huyo Citizen b.

Click to expand...

Hivi mimi nawewe nani anadanganya...umeniwekea proof za kuassume apa proof uchwala unaassume hesabu unazani hii ni physics

Citizen B said:

Hivi mimi nawewe nani anadanganya...umeniwekea proof za kuassume apa proof uchwala unaassume hesabu unazani hii ni physics

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Kwa ku-asume nilikuwaa nakupima na kukutania tu, lakini unapoleta jambo katika umma lazima uwe muangalifu sana kwasababu huwezijua back ground za watu wote humu jf, 500,000 members+,

Jibu la hiyo series ni lile nililokupa, kwasababu hiyo series ina progress arithmatically, hivyo Sum (S) yake ni hii;

S=[∞(∞+1)]/2, ili kujua kama ina converge au diverge, fanya ∞>>>1, hivyo ndani ya brackets ndogo ∞+1 becomes ∞, hence the whole equation becomes, S=(∞²)/2, sasa hapo utaona jinsi ∞² ilivyokuwa infinite number, pia kumbuka ∞/a=∞, where a is any positive integer number, in our case a=2, hivyo unapata;

S=(∞²)/2 diverges to infinity, kwasababu ya ∞².

Arithmetic progression na geometric progression ni topic za form 3.

Oya wanangu tunatiana Njaa

CONTROLA said:

Oya wanangu tunatiana Njaa

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Ndio safi ili ukapige sembe.

huu uzi nimekosea sana kuufungua, nlikua nmeanza kulewa pombe imekata, hapa naanza upya tena kudadeq, OP umefeli sana mkuu.

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Umenikumbusha documentary/ movie inayomuhusu, inaitwa 'The Man Who Knew Infinity'

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