- For
prove,
- For
, prove
-
Proof. Recall the Binomial theorem,
So, letting
and
we have,
-
Proof. First, we prove the left inequality,
If
, then
, so using the Binomial theorem we have,
Where we know the inequality is strict since there is at least one term (which is necessarily positive) in
since
.
Next, we prove the middle inequality,By part (a) we know,
Further, for
we have
for all
. Thus, we know that,
Hence,
for all
. Therefore, we have established the second inequality,
Finally, we prove the right inequality,
Here we expand the first few terms and use a previous result,
for all
. In the second to last line we used this result on the
th powers of a real number
(in this case
). This completes the proof for all of the inequalities requested
In the second inequality is false that (1 – r/n) < 1 for r = 0.
How did you reindex the product in part a?