Your work on part a) could a bit smoother: implies
.
For part b) consider .
Question:
We work in the real numbers. Are the following true or false? Give a proof or counterexample.
(a) If converges, then converges.
(b) If converges, then converges.
(c) If for all , and converges, then as .
(d) If , for all , and converges, then as .
(e) If is a decreasing sequence of positive numbers, and converges, then as .
MY WORK:
(a) and (b) can be proved similarly. Since converges, for some , when , then . Take s.t., . That is, . Also, . This implies that and therefore converges.
(c) This is where I get confused. THis seems like an indeterminate form, we have never done this in class. same for (d) and (e). any suggestions?
Here is part e).
For ease of notation use for the partial sums.
Because the series converges , the sequence is Cauchy.
If there is positive integer such if then .
We also know that . So there is a positive integer such that .
Now we have setup the machinery, we just have to find the tricks.
If then .
But we notice that
Can you finish?