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let-P-k-C-k-4-k-1-determine-C-so-that-the-family-p-k-k-N-define-a-measure-of-probability-2-we-draw-a-number-in-N-according-to-the-probability-P-determined-the-probability-that-the




Question Number 123100 by pticantor last updated on 23/Nov/20
let P_k =(C/(k×4^(k ) ))  1) determine C so that the family  (p_k ,k∈N^∗ ) define a measure of probability  2) we draw a number in N^∗  according to the   probability P determined the probability that the number  drawn is a multiple of 4           please i need your help
letPk=Ck×4k1)determineCsothatthefamily(pk,kN)defineameasureofprobability2)wedrawanumberinNaccordingtotheprobabilityPdeterminedtheprobabilitythatthenumberdrawnisamultipleof4pleaseineedyourhelp
Answered by Olaf last updated on 23/Nov/20
1)  Σ_(k=1) ^∞ P_k  = 1  CΣ_(k=1) ^∞ (1/(k4^k )) = 1  (1/(1−x)) = Σ_(k=0) ^∞ x^k   −ln∣1−x∣ = Σ_(k=0) ^∞ (x^(k+1) /(k+1)) = Σ_(k=1) ^∞ (x^k /k) if ∣x∣ < 1  If x = (1/4), −ln(3/4) = Σ_(k=1) ^∞ (1/(k4^k ))  ⇒ −C.ln(3/4) = 1 ⇒ C = (1/(ln(4/3)))  2)  P_k  = (1/(k4^k ln(4/3)))  Let k = 4j, j∈N^∗     Σ_(j=1) ^∞ P_k  = (1/(ln(4/3)))Σ_(j=1) ^∞ (1/(4j4^(4j) )) = (1/(4.ln(4/3)))Σ_(j=1) ^∞ (1/(j256^j ))  Σ_(j=1) ^∞ P_k  = (1/(4.ln(4/3)))Σ_(j=1) ^∞ (1/(j256^j )) = −((ln∣1−(1/(256))∣)/(4.ln(4/3)))  Σ_(j=1) ^∞ P_k  = ((ln((256)/(255)))/(4.ln(4/3)))  (I′m not so sure of my result)
1)k=1Pk=1Ck=11k4k=111x=k=0xkln1x=k=0xk+1k+1=k=1xkkifx<1Ifx=14,ln34=k=11k4kC.ln34=1C=1ln432)Pk=1k4kln43Letk=4j,jNj=1Pk=1ln43j=114j44j=14.ln43j=11j256jj=1Pk=14.ln43j=11j256j=ln112564.ln43j=1Pk=ln2562554.ln43(Imnotsosureofmyresult)

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