(1) Let {an} be a bounded sequence. How is lim sup an related to lim sup a2n and lim sup a2n+1?(2) The following are exercises that follow easily from definitions and theorems.(a) (Pringsheim) Let ∑an...

(1) Let {an} be a bounded sequence. How is lim sup an related to lim sup a2n and lim sup a2n+1?(2) The following are exercises that follow easily from definitions and theorems.(a) (Pringsheim) Let ∑an be a convergent series of positive decreasing terms. Prove nan → 0. Hint Use the Cauchy criterion. Deduce that the harmonic series is divergent.(b) If | ∑n j=1 a2j | ≤ M for all n and | ∑n j=1 a2j−1| ≤ M for all n then | ∑n j=1 aj | ≤ 2M for all n. (c) Show that if e ix ̸= 1 the geometric series ∑∞ n=1 e inx has bounded partial sums. Deduce that ∑∞ n=1 cos(nx) and ∑∞ n=1 sin(nx) have bounded partial sums. Use Dirichlet test to show ∑∞ n=1 e inx np is convergent provided e ix ̸= 1 and p > 0. (3) In class we did Abel’s classical summation by parts formula (N.H. Abel, Untersuchungen über die Reihe 1+mx+m(m−1)x 2+. . . , J. Reine Angew. Math. 1 (1826) 311–339). See https : //www.math.ust.hk/excalibur/v11n3.pdf for examples. (a) Use the Summation by Parts formula to find ∑n k=1 k2 k . (b) Prove Abel’s inequality: if m ≤ ∑n n=1 aj ≤ M for all n and {bn} is decreasing positive sequence then b1m ≤ ∑n k=1 akbk ≤ M b1. (c) Prove the following (more symmetric) version of the Summation by Parts formula: 1 2 ∑ q n=p an(bn+1 − bn) = aq+1bq+1 − apbp − ∑ q n=p bn+1(an+1 − an).(4) (Stolz–Cesàro theorem) Let {yn} be an increasing sequence whose limit is ∞. Let {xn} be a sequence. Prove that if { xn+1 − xn yn+1 − yn } is convergent then { xn yn } is convergent and lim ( xn+1 − xn yn+1 − yn ) = lim xn yn .(5) In the following exercise we will prove that 1 1 + 1 2 +. . .+ 1 n −ln n is convergent. Its limit, denoted by γ, is called Euler constant. (a) Prove that ln(1 + x) ≤ x, x ≥ 0. Hint. Consider g(x) = x − ln(1 + x). Then show g(0) = 0, g′ (x) ≥ 0. (b) By considering Riemann sums of ∫ n 1 1 x dx prove that 1 2 + 1 3 + . . . 1 n ≤ ln(n) ≤ 1 + 1 2 + 1 3 + . . . 1 n − 1 . Deduce that 1 + 1 2 + 1 3 + . . . + 1 n − 1 − ln(n) ≤ 1. (c) Prove 1 + 1 2 + 1 3 + . . . + 1 n − 1 − ln(n) is an increasing bounded sequence; hence, it is convergent. Its limit is the Euler constant γ. (d) From the above result (5c), we get 1 + 1 2 + 1 3 + . . . + 1 n − ln(n) isalso a convergent sequence whose limit is also the Euler constant γ. See the paper in the Proceedings of the American Math Soc., by Jonathan Sondow Dr. R. Attele 3
(6) By Cauchy Condensation test prove ∑n=3 1 n ln n ln (ln n ) is divergent but ∑n=3 1 n ln n (ln (ln n)) 2 is convergent.