Question

If you could please answer 8.1 & 8.2 that would be incredible. Thank you so much.

8.1. To show xn 2 Xn+1 for n 2 N and to find the value of N EN, consider the function. f: (0, +00) → R and so f(t) = n) Observe that if we restrict the domain of f to N then f(n) = n(n) = xn Therefore, if f is decreasing, then as will {xn} be decreasing. This suggests using methods from Calculus to determine the monotinicity of f. 8.2. Let {sn} be the real sequence defined negatively by si = 1 and for n 2 n + 1, Sn+1 = (1 + Sn). (a) Find S1,S2,S3,S4,S5. (b) Show that sn > į for all n e N. (c) Show that {sn} is a non-increasing sequence. (d) Prove that {sn} converges. (e) Find the value of limn+00 = Sn.

Answer 1

Senti = 3 ton) Then S2 = 3 Ctsi) = 7CH) z = f (H2) = 3 (1H2) = 1/2 Sy = $(133) = $Cho 55 = $(457) = $(14) ) = ③ we pore sn= 2 *nen by indention on new If nathen Sal> Ž & The result is freee formal Assume that result is true for not - that is sixth for KEIN S** - $ (1+Sx) > 5 6+5)=3-() => Skti> Hence The result is free for nanti. Then by induction onn, Snt & MEN

© Brom @ clesly six Sz 6;1>$) The result is tree fal nal Assume that result is there for n=k Them Sk> Sato Phat is ShHTSA Sit2 = $ CI+S2+)< $(+5) Skate =) SKHRESAH Hence result is free for nakti By induction on nen, ShHTn then hsnf- is decreasing sequence. that sint is non increasing sequeno ♡ sme tship is decreasing sequence. and bas ended below (asy Sn> Žrnent osemfis bocopted bookes by sinf Converges 18 let im Sn=d. then Im Stad

from Snte = 11t5n). hormon that a home (145) d = 3 Citd) 3d=HU zdal d=/ Honce imat