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Question Number 100624 by Dwaipayan Shikari last updated on 27/Jun/20
Find the value of (√(2+(√(2+(√(2+(√2)))))))...∞ using cos function
Findthevalueof2+2+2+2usingcosfunction
Commented by MJS last updated on 27/Jun/20
why cos?  x=(√(2(√(2(√(2(√(2....))))))))  x^2 =2x  x_1 =0 wrong  x_2 =2 is the solution
whycos?x=2222.x2=2xx1=0wrongx2=2isthesolution
Commented by Dwaipayan Shikari last updated on 27/Jun/20
There was a mistake on my question. Now can you prove it  sir using cos function????  I have edited my question
Therewasamistakeonmyquestion.Nowcanyouproveitsirusingcosfunction????Ihaveeditedmyquestion
Commented by Dwaipayan Shikari last updated on 27/Jun/20
Sir i have found something  1+cos(π/4)=2cos^2 (π/8)  (√((1/2)(1+(1/( (√2))))))=cos(π/8)  (1/2)(√(2+(√2)))=cos(π/8)=cos(π/2^3 )  1+cos(π/8)=2cos^2 (π/(16))  (1/2)(√(2+(√(2+(√2)))))=cos(π/(16))=cos(π/2^4 )  So  (1/2)(√(2+(√(2+(√(2+(√(2+(√(2+(√(2+(√(2...))))))))))))))n=cos(π/2^(n+1) )  lim_(n→∞)   (1/2)(√(2+(√(2+(√(2+(√(2+(√(2+(√2)))))))))))..n=cos(0)=1  so(√(2+(√(2+(√(2+(√(2+....∞))))))))=2  is it right???????
Sirihavefoundsomething1+cosπ4=2cos2π812(1+12)=cosπ8122+2=cosπ8=cosπ231+cosπ8=2cos2π16122+2+2=cosπ16=cosπ24So122+2+2+2+2+2+2n=cosπ2n+1limn122+2+2+2+2+2..n=cos(0)=1so2+2+2+2+.=2isitright???????
Answered by maths mind last updated on 27/Jun/20
u_(n+1) ^2 =u_n +2...E  u_0 =(√2)  we can proof this  withe  logic  0≤u_n ≤2...easy too see  u_(n+1) −u_n =((u_n +2−u_n ^2 )/( (√(u_n +2))+u_n ))=((−(u_n −2)(u_n +1))/( (√(u_n +2))+u_n ))≥0  u_n ↾ increasing   ⇒  since0 ≤U_n ≤2  ⇒∃ (w_n ) such u_n =2cos(w_n ) proof  w_n =cos^(−1) ((u_n /2)) well defind since (u_n /2)∈[0,1]  since u_n ∈[0,2] and incrasing ⇒w_n ∈[0,(π/2)]⇒  w_n  is deacrising  E⇔4cos^2 (w_(n+1) )=2+2cos(w_n )⇒2cos^2 (w_(n+1) )−1=cos(w_n )  cos(2x)=2cos^2 (x)−1⇒  E⇔cos(2w_(n+1) )=cos(w_n )  ⇒2w_(n+1) =w_n  because w_n ∈[0,(π/2)]  ⇒w_(n+1) =(w_n /2)⇒w_n is geometric sequences  ⇒w_n =w_0 ((1/2))^n   w_0 =(π/4)  u_n =2cos((π/(4.2^n )))=2cos((π/2^(n+2) ))
un+12=un+2Eu0=2wecanproofthiswithelogic0un2easytooseeun+1un=un+2un2un+2+un=(un2)(un+1)un+2+un0unincreasingsince0Un2(wn)suchun=2cos(wn)proofwn=cos1(un2)welldefindsinceun2[0,1]sinceun[0,2]andincrasingwn[0,π2]wnisdeacrisingE4cos2(wn+1)=2+2cos(wn)2cos2(wn+1)1=cos(wn)cos(2x)=2cos2(x)1Ecos(2wn+1)=cos(wn)2wn+1=wnbecausewn[0,π2]wn+1=wn2wnisgeometricsequenceswn=w0(12)nw0=π4un=2cos(π4.2n)=2cos(π2n+2)
Commented by ghiniboss last updated on 28/Jun/20
what aspect of maths is this?
whataspectofmathsisthis?
Commented by prakash jain last updated on 28/Jun/20
Two topics  Sequences and series  recurrence relation
TwotopicsSequencesandseriesrecurrencerelation

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