Consider two sequences and
whose terms are always positive after some integer
(i.e.,
and
for all
). Then define
Prove the following two statements.
- If there exists a constant
with
for all
, then
converges.
- If the series
diverges and if
for all
then the series
diverges.
- Proof. First, we solve the given relation for
(starting with the term
so that all terms are positive, hence, nonzero),
Then, since
for all
, we have
Therefore, the partial sums of
are bounded (since
is a constant); hence,
converges
- Proof. We are given
which implies
This then implies
Since
is a constant this implies that if
diverges so does
Here is my proof for part a).
we know from the problem statement
multiplying by
and dividing by
we get
summing up such inequalities for
to
we have
the first sum collapses by the telescopic property
since
we have
the sequence of partial sums of
is thus bounded. And since it’s increasing too (
), then it must converge by theorem 10.1.
I solved it the same way! I will add my proof of (b) :
$$b_n-\frac {b_{n+1}a_{n+1}}{a_n}≤0$$ $$a_nb_n≤b_{n+1}a_{n+1}$$
$$0≤a_Nb_N≤a_nb_n$$
$$0≤\frac {a_Nb_N}{b_n}≤a_n$$
Just noticed a couple of typos. See suggestions below.
In the proof for part a:
for all
, we have
\begin{frame}
Then, since
\end{frame}
In the proof for part b:
\begin{frame}
\end{frame}
In the proof for part a, should the sentence “Then, since 0 < c_N \leq r for all n \geq N, we have" be "Then, since 0 < c_N \leq r for all n \geq N, we have" instead?
In the proof for part b,