Advanced Calculus fi..

Chapter 6 Infinite Series 383PROBLEMS1. Show that the following sequences converge and find their limits:> .:qd) n log (1 + k), e) s,=1 forn=1,2,3 ,... a. t .2. Determine the upper and lower limits of the sequences:a) cos nn, b) sin f nn, C) n sin ~ 1 nn.3. Construct sequences having the properties stated:-C) limn,oosn = +oo, lim, s, = +oo.4. Show that the Cauchy criterion is satisfied for the sequence s, = lln.5. Evaluate e to 2 decimal places from its definition as limit of the sequence (1 + lln)".-6. Evaluate lim,,,xn and lim,,,xn.7. The number n can be defined as the limit, as n + oo, of the area of a regular polygon of2" sides inscribed in a circle of radius 1. Show that the sequence is monotone and use itto evaluate n approximately.IAn infinite series is an indicated sum:of the members of a sequence a,. The series can be abbreviated by the C sign:sdFand the notation is to be preferred, except for very simple series.Associated with each infinite series C a, is the sequence of partial sums S,:n=l(Note that the index j in the sum here is a dummy index, so that the result doesnot depend on j. Thus C;=, a, = C;=, a,. This is like the case of the integrationvariable in a definite integral.) Accordingly,S1 =a!,S2=al+a2, ... , Sn =al +..-+a,.DEFINITION The infinite series Czl a, is convergent if the sequence of partialsums is convergent; the series is divergent if the sequence of partial sums is divergent.If the series is convergent and the sequence of partial sums S, converges to S, thenS is called the sum of the series, and one writes