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Berkeley MATH 1B - Midterm review

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Math 1B midterm review, Spring 2002Yosen Lin and Karen KapurApril 9, 2002Series solutionsImportant concepts1. Sequences. Convergence and divergence. Monotonic sequences.2. Series testsSingle series testsIntegral test. The function must be positive and decreasing.Divergence test.Alternating series.Absolute convergence testing. Note that absolute convergence does not imply conditional convergence.Root and ratio test.Comparison testsSums and multiples of series. Be careful - the difference of two diverging series may be convergent.Direct comparison test. All terms must be positive.Limit comparison test. All terms must be positive.Tests with a known remainderIntegral test.Direct comparison test.Alternating series.Useful series to knowArithmetic series.Geometric series.Telescoping series.Harmonic series.Alternating harmonic series.p-series.Binomial series.3. Series heuristics(a) Is it one of the above series? See if the series is one of the above series, or if it can be manip-ulated (using some algebra) into one of the above forms. If this is possible, then simply apply thecorresponding formula for convergence.1(b) Divergence test. The divergence is usually easy to apply. It will tell immediately if a series diverges,and can usually be done in less than a minute.(c) Comparison with p-series. If the terms of the series are a ratio of polynomials, then often acomparison with p-series will be effective. Try either the limit comparison or the direct comparison.A good rule of thumb is to use the difference of the highest powers in the numerator and denominatoras the value for p).(d) Root and ratio test. Any series with an expression to an n-th power may be a candidate for theroot test. Such series are also candidates for the ratio test, as well as any series involving factorialsand products. (Using the ratio test will often eliminate factorials in the resulting expression.)(e) Comparison tests. If a series is close to, but not exactly the same as, a standard series, a comparisontest is a good one to try. Remember the following two standard limits; they may be helpful in a limitcomparison test:limx→0sin xx= 1limx→0cos x − 1x= 0(f) Absolute convergence testing. If a series has both positive and negative terms, but the signs switcharbitrarily, then the test for absolute convergence must be used. Test to see if the corresponding series(of absolute values of the original series) converges or diverges.(g) Alternating series test. A series with alternating positive and negative is a candidate for thealternating series test. Be sure to verify that the series satisfies the hypotheses for the alternatingseries test. Don’t be fooled by series that don’t look like alternating series, but really are. For example,does∞Xn=1cos(nπ)(1n)converge or diverge?(h) Integral test. If it looks like the infinite integral of the series can be evaluated, then the integraltest is a possible choice. Be sure the series satisfies the hypotheses for the integral test. Usually a lastchoice, as integration is tedious in general, as is verifying the integral test hypotheses.4. Power series. Taylor and Maclaurin series.Power series and radius and interval of convergenceTaylor and Maclaurin series for arbitrary functionsFinding the sum of a power seriesBasic ProblemsDetermine the convergence or divergence of the series.1.P∞n=11e2n2.P∞n=1ne−n23.P∞n=13n2n4.P∞n=1n2+1n4+15.P∞n=1(−1)n−1√n6.P∞n=1(−1)n+1n2n7.P∞n=1(−1)n5+n8.P∞n=11(2n)!2Find the radius of convergence and the interval of convergence.9.P∞n=0xnn!10.P∞n=1xnn3nFind a series expansion (centered at 0) for the following functions.11. f(x) =x1−x12. f(x) = cos πx 13. f(x) = x2e−x14. f(x) =R11+x4dx 15. f(x) =Rsin xxdxAdvanced Problems1.P∞n=1sin(1n) 2.P∞n=1e−nn! 3.P∞n=1n2e−n34.P∞n=1n!nn5.P∞n=1tan−1nn√n6.P∞n=1(−1)nnnn!7.P∞n=1n+53√n7+n28.P∞n=1(−1)nsin(πn)9.P∞n=11n1+1/nFind series expansions for the following functions (centered at 0).10. f(x) = x ln(1 + x) 11. f (x) = x arctan(x3)12. f(x) = ln(9 + x2) 13. f(x) = sin2x14. f(x) = sin(x4) 15. f(x) =Rx0sin(t2) dtFind a closed formula for the sum.16.P∞n=0(−1)nπ2n+142n+1(2n+1)!17.P∞n=0(−1)nxn22nn!18.P∞n=0(x+2)n(n+3)!Differential EquationsImportant concepts1. Techniques for solving differential equationsSeparation of variablesIntegrating factors (Linear equations)3Basic ProblemsSolve the differential equation.1.dudt= 2 + 2u + t + tu 2. y0+ 2y = 2ex3. 1 + xy = xy04. y0= x2y5. (1 + t)dudt+ u = 1 + t 6. xyy0= ln x, y(1) = 2MiscellaneousShowing a sequence is increasing or decreasingThere are three ways to show that a sequence is increasing. Some of them are easier to apply in certain situationsthan others. The first test assumes that all elements are positive:Division Ifan+1an≥ 1 then the sequence is increasing. This follows from dividing both sides of an+1≥ an.Subtraction If an+1− an≥ 0 then the sequence is increasing. This follows from subtracting both sides ofan+1≥ an.Differentiation If f0(n) ≥ 0 then the sequence is increasing. This follows from the definition of derivative.There are similar formulas to show that a sequence is


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Berkeley MATH 1B - Midterm review

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