Examples of convergent sequences - exercises

Prove that the sequence ( ₎ converges to zero when is given by:

(a) (-1) 1 ; (b) ²⁺² ; (c) ; (d) +1 ; (e) (+1)³ - ^{3+3 2} .

(In (c), state precisely what properties of the sine function you are using.)

Let _{= +1} for =1,2,3, .

State what it means for ₁ as and carefully prove this statement.

Stating results from the text where appropriate, show that ₌ defines a convergent sequence for =0 or =1 and a bounded by divergent sequence for =-1.

Let and let ₌ .

(a) Writing =1+ and using Bernoulli's inequality, show that the sequence ( ₎ does not converge to any real limit when 1 .

(b) Show that the sequence ( ₎ does not converge to any real limit when -1 .

(Standard results from the course may be used if quoted correctly. In particular, if you don't want to prove the results directly, the idea of a bounded sequence may be helpful.)