Math 312 S08 Day 2
1. Proposition 9.5. Suppose that the series
!
"a
k =1
k
converges. Then the
sequence {ak} must converge to zero.
Proof: The trick to this proof is to first note that the difference between
the nth term (sn) and the n-1st term (sn-1) of the sequence of partial sums is
the nth term of the original sequence, {an}.
Once we note that, we cleverly think of {sn} and {sn-1} as two different
sequences. Of course {sn-1} is just the first n-1 terms added together
rather than the first n terms. So, it is really just the same sequence with a
zero at the beginning. Here goes the proof…
By definition of convergence for series, the sequence of partial sums
{sn} must converge and so must the sequence {sn-1}. Clearly these two
sequences must converge to the same number. Thus by the difference
property of convergence, we know that lim (sn # sn#1 ) = lim sn - lim sn#1 = 0.
n!"
n!"
n!"
But of course sn ! sn!1 = an. So we have lim an =0 as was claimed.
n!"
Group Work (you should not need the book for these):
!
1. Now we know that in order for a series
"a
k =1
k
to converge, the sequence {ak}
must converge to zero. This is what we call a necessary condition. But is it a
sufficient condition?
!
Consider the series
1
" k . This is called the harmonic series.
k =1
a. Review problem: Use the definition of convergence for sequences to
!1 $
prove that the sequence " % converges to zero.
#k &
b. Let {sn} be the sequence of partial sums for the harmonic series.
n
Prove that for each n, s2n ! 1 + . Use this fact to prove that the
2
harmonic series diverges.
Now we know that the condition that {ak} converges to zero is NOT sufficient to
!
ensure that the series
"a
k =1
k
converges.
2. Suppose that you have a sequence of non-negative numbers {ak}.
a. What can you say about the sequence of partial sums of the series
!
"a
k =1
k
?
b. Complete (and prove) the theorem:
!
Suppose {ak} is a sequence of non-negative numbers. Then
"a
k =1
k
converges if the
sequence of partial sums {sk} is ____________.
3. Suppose that you have two sequences of non-negative numbers {ak} and {bk}.
Suppose also that for each k, ak ≤ bk. Is there a connection between the
!
convergence/divergence of
" ak and
k =1
!
"b
k =1
k
? Justify your conjectures.
4. Examine the following series for convergence:
!
a.
5
"k
k =1
!
b.
1
" 5k
k =1
!
5. Suppose that you have a sequence {ak}. The sum of the series
"a
k =1
k
is really
just the result of adding up all of the terms of the sequence. Right?
We all know that addition of real numbers is associative, so we can group
numbers however we want before adding them. In particular, given the numbers
a, b, c, and d, we know that a + b + c + d = (a + b) + (c + d).
!
a. Suppose we did this same thing with the series
"a
k =1
k
(group the terms
into pairs before adding them). How would we express this new sum
using series notation?
b. Will this new sum always have the same limit as the original? Does it
converge if and only if the original converges?