Calculus III Fake Quiz 1 Spring 2000 1/19/2000


1. Determine whether the sequence  converges or diverges. If it converges, give the limit and show how you know it converges to that value. Give justifications for each step in your reasoning.

The sequence converges to 1.

We start out by using the standard trick of dividing each term by the highest power of n in the expression, i.e. n2, so .

Then we note that (by the limit laws listed on p. 601 if we're very picky about it) the limit of this sequence is equal to the limits of the individual terms, so the previous limit is equal to .

We know the limit of the constant 1 is 1 (again by the limit laws on p. 601 if we're super-picky)
and we know first that the limit of 1/x is 0 when 1/x is a function and second that this fact carries over to sequences (by a Theorem from section 3.5 and Theorem 2 on p. 600, respectively), so our limit simplifies to =1.
 
 

2. Determine whether the sequence an converges or diverges. If it converges, give the limit and show how you know it converges to that value. Give justifications for each step in your reasoning.

The sequence converges to 0.

We start out by noting that cos n is bounded by -1 and 1, so , and the same inequality applies to the respective limits.

Next note that , where we have used the limit laws from p. 601, the fact that the limits of 1/x and 1/x2 are both 0, the recent Theorem saying that this carries over to the sequences 1/n and 1/n2, and assorted basic properties of algebra.

The same can be done for the expression on the right end of our inequality, so we have our sequence trapped between two sequences which both converge to 0. This exactly the situation required to apply the Sqeeze Theorem for Sequences (from p. 601, or alternately note that it works this way for functions and carries over to sequences) so our sequence must converge to 0 too.
 
 

3. Find the first four partial sums of the series .
 

s1 = 1/1 = 1

s2 = 1/1 + 1/2 = 3/2

s3 = 1/1 + 1/2 + 1/3 = 11/6

s4 = 1/1 + 1/2 + 1/3 + 1/4 = 25/12
 

There's not a lot of pattern to the successive partial sums, and you might now recognize this as the harmonic series which in fact diverges to infinity.