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Review sheet for Mathematics Exam 1

Main concepts:

Partial fractions: - Remember that: A prime (i.e. indecomposable) polynomial (with real coefficients ) can only
be either a degree one polynomial or a quadratic that has no real zeros (i.e. the discriminant b2-4ac < 0). Any
polynomial of degree 3 or higher can still be factored.

- The simple case : all factors in the denominator are linear , and no factors are repeated. The good thing about
this case is that you can evaluate the constants quite easily by evaluating the rational function at certain values
of x.

- Second case: Factors in the denominator are all linear but some are repeated. The decomposition scheme into
simpler fractions is slightly changed. Evaluation of the constants involve substituting certain (special) values of
x and then trying
to get as many equations as the unknowns .

- Third case: Factors in the denominator are linear or prime quadratic. Some factors are repeated. Very similar
to the second case. Just be careful that when you break it down into simpler fractions, the numerator part
corresponding to a prime quadratic has to be on the form Ax+B.

Essential skills - Factoring polynomials completely into prime factors.

- Long division of polynomials can be indispensable for some problems.

- Solving linear equations in several unknowns by eliminating the unknowns one at a time.

- Being able to integrate the the simpler fractions, specially factors with a prime quadratic in the denominator.
Recall that you may have to complete the square in order to be able to do that.

Sequences: - One very important thing here is to be able to distinguish between a sequence and a series. A
sequence is a function in n, while a series is an infinite sum of the terms of a certain sequence.

- Limit problems for sequences do not involve anything new, they are very similar to limit problems from
Math
121.

- Some times you have to prove the existence of the limit of a sequence before trying to evaluate it, say, by using
a recurrence formula. To prove the exitence of the limit, you may use the theorem we covered in class saying
that a monotone and bounded sequence has to have a limit.

Essential skills - All the skills and tricks you learned in evaluating limits in Math121 are relevant. Namely,
algebra of limits , L’Hopital’ s rule , the logarithmic trick, dividing both numerator and denominator by the highest
power of n...etc.

- Being able to show that a given sequence is (increasing) non-decreasing or (decreasing) non-increasing
by comparing to .

Infinite series: - Again, an infinite series, , is an infinite sum of terms of a certain sequence. Convergence
of a series is defined in terms of the convergence of the sequence of partial sums

- Two important examples of series that we know their behavior very well are the geometric series and telescoping
series. They are among some of the few series that we know their sum in case they are convergent.

- The nth-term divergence test applies to any series, and can only be used to prove that a series diverges. You
can NOT use it to prove convergence.

- The absolute convergence test, which states that a series that is absolutely convergent is also convergent,
can be applied to any series. But you can only use it to prove convergence. You can NOT use it to prove
divergence. There are countless examples of series that are convergent but not absolutely convergent. In these
cases, the convergence is due to the fact that the negative terms cancel out with the positive terms barely evading
divergence.

- Series with positive terms: Infinite series with positive terms are quite special since their sequence of partial
sums is always increasing (as you keep adding positive terms). By the theorem on monotone sequences, to prove
that a series with positive tems is convergent, it’s enough to show that its sequence of partial sums is bounded
from above.

Because of this property , there are several tools (tests) we can use to study their convergence or divergence.
Namely, the integral test, pth-power test, the direct and limit comparison tests, and the ratio test.
Intuitively, the convergence of a series with positive terms is just a matter of how fast the nth-term of the series
approches zero as n → ∞.

- Alternating series: - The alternating series test can be used to prove convergence of such series. It can never
be used to prove divergence.

- Power series: - Power series is a series that depends on a parameter (or a variable), say x, and it is on the
form . Each time you substitute a real number for x, you get a numeric infinite series. The most
important question is to find all the values of x that yield a convergent series. The main theorem on power series
paints a very specific picture of what goes on (check your class notes). We can then use the ratio test to test for
absolute convergence and determine the radius and the interval of convergence.

- Using the formula for the sum of a geometric series, and with some clever algebraic manipulation, we can
ex press some functions as the sum of a power series about a given center. This is a good thing to have, and we
will explore this further later in chapter 8.

- Error estimates: These are essential for numerous real life applications of infinite series. So far we had two
estimates, the integral test estimate, and the alternating series estimate, check your notes.

Essential skills
- What to do to investigate the convergence or divergence of a, numeric, series?
In a nut shell: Apply the nth-term divergence test. If it fails, apply the absolute convergence test, and then
you could use all the tests for series with positive terms. If the absolute convergence test fails, try to see if the
series is alternating and if the alternating series test applies proving convergence. For the problems you’ll see
this semester, the above plan should always work. If it doesn’t, then it simply means that you made a mistake
somewhere, retrace your steps and correct it.

- You have to know how to apply ALL the tests we covered so far. You have to have a good sense of approximating
the nth term of a series with positive terms in order to gauge how fast it goes to zero. Then use any of
the comparison tests to prove it.

- When applying the alternating series test, you may have to use some Math121 skills to show that the positive
part of the nth term is decreasing, e.g. as in .

Problems: The most important problems to review are, in this specific order, your homework problems, examples
done in class, and problems on the quizzes. The following set of problems is a just collection of particularly
tough questions to serve the purpose of reviewing as many concepts as possible. They do not necessarily represent
the questions you will see on the exam. The problems on the exam will be much closer to the ones from
your homework, or the ones we did in class than the ones found here.

Good luck, see you on Monday if you have any questions.

(1) Evaluate the following limits.

(2) Which of the following series converge and which diverge? Justify your answer by naming the tests you are using.

(3) Find the interval and radius of convergence for each of the following power series. Justify your answer

(4) Find the value a for which the following series converges to 2/3 .

(Ans. -ln2)

(Food for thought: Could we ask the same question and have the sum be equal to   instead of ?)

(5) Is it possible to write the function as the sum of a power series about x=0 ? Explain your answer,
and find the values of x that make this possible?

(6) Expand each of the following quotients by partial fractions

(Answer: ) (Answer: A=3 , B =14, C =-3 , D =-7 )
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