**Question 1.**
A common mistake was to state that if Ratio test is inconclusive,
convergence is not absolute. This is not right - Ratio test isn't
necessary to detect absolute convergence. Absolute convergence
simply means that the series converges when you replace its terms
with their absolute values. You can then study the series of absolute
values by any method. The Ratio test considers absolute values
(because of the way it's set up), so a convergence outcome of the
Ratio test automatically implies absolute convergence. But the
failure of the Ratio test
doesn't imply anything.

(a) Answer: the series is not absolutely convergent because the series of absolute values is a p-series
with p=2/3. However, the series is conditionally convergent by the Alternating
Series Test. When applying the Alternating Series Test, you need to explain
why the sequence is decreasing (this must involve ALL terms, not just
saying that b_{2}≤ b_{1}. (See also comment to Question 2).

(b) The series is convergent by the Comparison Test.
Since the terms are all positive, convergence also implies
absolute convergence. Few people stated that the terms were positive;
some omitted checking convergence of the series they used for comparison.

(c) Many students made mistakes simplifying the expression
(esp. factorials) in the Ratio test.

** Question 2. **

(a) The sequence converges to 1/2. For full credit, you had to include
the algebra justifying the limit calculation. Unfortunately, a number of students
also confused convergence of sequence with convergence of series.

(b) The sequence is bounded below by 0 (because the terms are positive)
and above by 1 (because the numerator is always smaller than the denominator,
thus the quotient is less than 1). There is no need to find best possible upper/lower
bounds.

There was a lot of confusion on this question -- many students attempted to study the sequence
for increasing/decreasing instead of upper/lower bounds. Many incorrectly claimed that
the sequence decreases (checking that the first term is greater than the second), and then
concluded that the sequence is decreasing from 2/3 (the first term) to 1/2 (the limit),
with 2/3 and 1/2 the upper (resp. lower) bounds. In fact, the sequence does the following:
it decreases for the first three terms (2/3, 1/2, 10/21 - note that 10/21 is less 1/2),
then it increases for the rest of the sequence, to the limit 1/2. To detect this, you'd have to
study the entire sequence, not just a couple of first terms -- but this was not required in the
question.

(c) Many people computed a_{1} and a_{2} instead of s_{1}
and s_{2}. Remember that s_{n} is a partial SUM - thus
s_{2} = a_{1}+a_{2}. Misinterpreting the question this way cost you
3 points. (By constrast, if you know what partial sums are but make a mistake in calculations,
you only lose 1 pt). Partial sums are defined for any series and generally have nothing to do with
particular formulas valid for geometric series.

(d) The series diverges by Divergence test.

(e) The sequence of partial sums is bounded below by 0, increases, and is not bounded above.
This is because partial sums are sums of positive terms of the series - you keep adding more
and more terms, so the sequence increases. The series diverges, this s_{n} goes to
infinity, and so cannot be bounded above. (Even with correct answers, you had to include
explanation to receive full credit.)

(f) Same as (d). Alternating test is irrelevant, and in fact doesn't apply since the
sequence is NOT decreasing. The failure of Alternating test doesn't imply any conclusion -
you really need to say that the terms of the series do not go to 0, which implies
divergence.

** Question 3. **

For questions about the radius/interval of convergence for power series, use Ratio Test.
(A number of people tried to argue that the given series was geometric, but it wasn't.)
When the Ratio test gives lim=1, the test is "inconclusive", which means just that --
you can't make any conclusions. A few students incorrectly claimed that lim=1 implies divergence.
Note that the Ratio Test NEVER works on endpoints of the interval of convergence
(those need to be checked by
other tests).

After you found the interval of convergence, there is no need to test points that are inside/outside the interval.
However, endpoints need to be tested separately.

** Question 4. **

(a) This series is divergent. Many students represented it as a sum of two geometric series, one divergent,
one convergent. However, for full credit you need to justify why convergent+divergent=divergent series.
An easier solution is to apply the Divergence test. Indeed, the first summand in each term oscillates between
large positive and large negative numbers, and the second approaches 0 as *n* grows large.

(b) Convergent series. Represent as a sum of two convergent geometric series to justify convergence and find the sum.

(c) This is also a geometric series, but one that depends on x. You can use the geometric series argument to find where the series converges.
Alternatively, you can use the Ratio Test, but for full credit (on "where converges" part) you need to check endpoints separately.
The Ratio Test doesn't help to find the sum (for this part, you have to use geometric series).