Review_exam1
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Review sheet for Exam1
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 Math121. - Sometimes 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 {an } is (increasing) non-decreasing or (decreasing) non-increasing by comparing an to an+1 .
∞
• Infinite series: - Again, an infinite series,
nБайду номын сангаас1
an , is an infinite sum of terms of a certain sequence. Convergence
N
of a series is defined in terms of the convergence of the sequence of partial sums SN =
n=1
an = a1 + a2 + . . . + aN
- 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
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 Math121. - Sometimes 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 {an } is (increasing) non-decreasing or (decreasing) non-increasing by comparing an to an+1 .
∞
• Infinite series: - Again, an infinite series,
nБайду номын сангаас1
an , is an infinite sum of terms of a certain sequence. Convergence
N
of a series is defined in terms of the convergence of the sequence of partial sums SN =
n=1
an = a1 + a2 + . . . + aN
- 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
