Citizen B
JF-Expert Member
- May 13, 2019
- 6,677
- 9,035
Good day good citizens...
Albert Einstein alisema na'quote
"Pure mathematics is, in its way, the poetry of logical ideas."
— Albert Einstein, German theoretical physicist
Leo tucheki hii mathemagic na poetry iliyo ndani ya infinite series; yani sequence zisizo na mwisho.
Tuanze kwa kujua series ni nini:
Series ni summation au sum ya terms Kwenye infinite sequence of numbers
Mfano: S=a1+a2+a3+a4+a5...∞
Yani ni lile jawabu utakalopata ukijumulisha sequence ya numbers mpaka infinity
Ulishawahi kujiuliza ukijumlisha 1+1+1+1+1+1+1+1....mpaka milele utapata jibu gani? Najua unahisi litakuwa jibu kubwa sana lakini mahesabu yanatuonesha ina kuja kuwa ka namba kadogo sana..lakini hiyo ni mada ya siku nyingine.
Leo tunaangalia series aina mbili ambazo ni convergent na divergent
Convergent series: hizi ni series ambazo kadri inapozidi kuendelea inazidi kukaribia namba fulani (tunaweza kuiita limit),hivyo hizi series tunaweza kuapproximate jawabu la mwisho
Mfano;1-1+1-1+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
Divergent series: hizi ni zile ambazo jawabu lake halikaribii namba yoyote kadri unapozidi kujumlisha terms zake
Mfano: reciprocal ya positive integers inaleta divergent series
1/1 + 1/2 + 1/3 + 1/4 + 1/5....∞(harmonic series)
Hii ni kwasababu
Ukichukua term ya tatu na nne i.e 1/3 +1/4 jibu litakuja kuwa kubwa zaidi ya 1/2
Lakini pia ukichukua term ya tano,sita,saba,nane jibu pia linakuja kubwa zaidi ya nusu i.e
1/5 + 1/6 + 1/7 +1/8 >1/2
Vivyo hivo kwa sum nane zitazofata zotakuwa kubwa zaidi ya 1/2
Hivyo utaona kadri unapozidi kwenda namba zinaongezeka na tukiapproximate Ziwe sawa na nusu italeta 1+ 1/2 +1/2 + 1/2...∞
Hivyo jibu halisogelei namba yoyote inaitwa divergent series..
Baada ya kuona aina za series Sasa swali linakuja kuwa je! Ukijumlisha 1+2+3+4+5+6+8+9...∞ ni convergent au divergent??
Jibu ni kuwa hiyo series ni convergent na inaleta jibu la -1/12
Yaani ukijumlisha 1+2+3+4+5+6+7 ....milele mpaka infinity ∞ utapata jibu la -1/12
That is to say
1+2+3+4+5+6+....∞ = -1/12
Kivipi?? Nitaprove hilo kwa Uzi unaokuja
Albert Einstein alisema na'quote
"Pure mathematics is, in its way, the poetry of logical ideas."
— Albert Einstein, German theoretical physicist
Leo tucheki hii mathemagic na poetry iliyo ndani ya infinite series; yani sequence zisizo na mwisho.
Tuanze kwa kujua series ni nini:
Series ni summation au sum ya terms Kwenye infinite sequence of numbers
Mfano: S=a1+a2+a3+a4+a5...∞
Yani ni lile jawabu utakalopata ukijumulisha sequence ya numbers mpaka infinity
Ulishawahi kujiuliza ukijumlisha 1+1+1+1+1+1+1+1....mpaka milele utapata jibu gani? Najua unahisi litakuwa jibu kubwa sana lakini mahesabu yanatuonesha ina kuja kuwa ka namba kadogo sana..lakini hiyo ni mada ya siku nyingine.
Leo tunaangalia series aina mbili ambazo ni convergent na divergent
Convergent series: hizi ni series ambazo kadri inapozidi kuendelea inazidi kukaribia namba fulani (tunaweza kuiita limit),hivyo hizi series tunaweza kuapproximate jawabu la mwisho
Mfano;1-1+1-1+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
Divergent series: hizi ni zile ambazo jawabu lake halikaribii namba yoyote kadri unapozidi kujumlisha terms zake
Mfano: reciprocal ya positive integers inaleta divergent series
1/1 + 1/2 + 1/3 + 1/4 + 1/5....∞(harmonic series)
Hii ni kwasababu
Ukichukua term ya tatu na nne i.e 1/3 +1/4 jibu litakuja kuwa kubwa zaidi ya 1/2
Lakini pia ukichukua term ya tano,sita,saba,nane jibu pia linakuja kubwa zaidi ya nusu i.e
1/5 + 1/6 + 1/7 +1/8 >1/2
Vivyo hivo kwa sum nane zitazofata zotakuwa kubwa zaidi ya 1/2
Hivyo utaona kadri unapozidi kwenda namba zinaongezeka na tukiapproximate Ziwe sawa na nusu italeta 1+ 1/2 +1/2 + 1/2...∞
Hivyo jibu halisogelei namba yoyote inaitwa divergent series..
Baada ya kuona aina za series Sasa swali linakuja kuwa je! Ukijumlisha 1+2+3+4+5+6+8+9...∞ ni convergent au divergent??
Jibu ni kuwa hiyo series ni convergent na inaleta jibu la -1/12
Yaani ukijumlisha 1+2+3+4+5+6+7 ....milele mpaka infinity ∞ utapata jibu la -1/12
That is to say
1+2+3+4+5+6+....∞ = -1/12
Kivipi?? Nitaprove hilo kwa Uzi unaokuja
