Question

Let (an) be a sequence such that lim an = 0. Define the sequence (AR) Exercise 21: by A =ļa, and An = zou-a + ax=a + zam for k21. Prove that an converges to some S if and only if Ax converges to S. N=0 k=0 Exercise 22: (Cauchy condensation test) Let (an) be a sequence such that 0 < antı san a) Show n=0 n=1 Hint: Recall the proof of convergence of for p > 1. Ren for b) Show that the series an converges if and only if the series 2"agn converges. 70 c) Deduce from part b) that converges for p >1 and diverges for p<1.

Answer 1

And 22) o ant £an Civen Ox an 3 n A2493 45 ) 3 1 +(ak)+ kzic 31 on. ( (A 4a4+ 893 + 16G16+ 32432 21+ 2 a2 32a52 (a,)+ 2a,)+ (4 a) 1G46 Now 2. azn +

(a,a) + (2a) + (2a4+ 2a4) + (2a,1 2a, 429 29) 2a10 t 29IC t +241 3 tim (2 a2nt 2 42t 2a1n 2n-1 times (2 4,)(2)(2 a3 + 244) 24, + 2 +292) 241) 2-t 2 a1 291 - 2 9n 2 ハニ」 2 2na2 -) Converj a2n tho

than n conveha Camboraon test) an Convena nl than 2 2 n n o M 2 2n Conveegis n o CCombrison) test b) NA n Σ 2 (ay6)) n-1 h 2 Conve ConveghA nb 22) sdenr, Cnvey frooved Hercs -E MA