solutions for c

# It will give you answer 2c 3 1 4 1 5 8 7 3 from

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Unformatted text preview: 2−1/13−1/14)/16+ (4/17−2/20−1/21−1/22) /(16 ∧ 2) ENTER in your calculator. It will give you Answer (2c) : 3 . 1 4 1 5 8 7 3 ... From calculator, π = 3.1415926535.. . The diﬀerence between the above answer in (2) and π is 0.000005... . It is between 10−5 and 10−6 . 3 . [III] (70pts) (1) Let s and λ be real numbers such that 0 < s < 1 and +∞ λ > 0. us−1 u+λ Evaluate u=0 , du as follows. +∞ Step 1. Agree that, for e−(u+λ) t dt u > 0, 1 . u+λ = t=0 ⋆ Just check the box ‘Agreed’. Step 2. +∞ u=0 Accordingly, us−1 u+λ +∞ u=0 +∞ = t=0 +∞ us−1 · du = e−(u+λ) t dt du t=0 +∞ u=0 us−1 · e−u t du e−λ t dt s Γ ts +∞ =Γ · s −s t · e−λ t dt t=0 Γ 1−s λ1−s =Γ ·Γ s 4 1−s · λs−1 . Taking into account the functional identity : Γs Γ 1−s π = sin , πs we conclude +∞ u=0 (2) us−1 u+λ π du = sin · λs−1 . πs We may rewrite the result of (1) as +∞ u=0 1 u+λ sin · us−1 · πs Integrate the both sides as in indeﬁnite integrals +∞ u=0 1 u+λ · us−1 · u=0 1 u...
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## This note was uploaded on 01/12/2014 for the course MATH 116 taught by Professor Scholle,minho during the Fall '08 term at Kansas.

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