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Diﬀerential Equations and Matrix Algebra II (MA 222), Winter Quarter, 1999-2000 Material to know from inÞnite series of numbers and Taylor series 1. The name of the inÞnite series ∞ 1 is k Is this inÞnite series convergent or divergent? k=1 2. This inÞnite series is convergent and ∞ 1 k=0 3. The name of the inÞnite series 2k ∞ (−1)k+1 k=1 = . Is this inÞnite series convergent or divergent? k ∞ 1 .Determine whether the k(k − 1) k=2 inÞnite series is convergent or divergent. If convergent. what is the value? 4. Be able to Þnd the partial sums of the inÞnite series ∞ 2n − 1 5. If ak = , then ak is convergent or divergent? n k=1 k=1 n 6. Given that the partial sums of the inÞnite series ∞ k=1 ak are sn = −2 + n+1 n (or some other expression), determine whether the inÞnite series is convergent or divergent. If convergent, what is the value? 7. Know that the inÞnite series ∞ 1 k=1 8. The name of the inÞnite series kα ∞ n=0 is convergent for α > 1 and divergent for α 1. xn . This inÞnite series is convergent or divergent.for which values of x? If convergent, what is its value? 9. Determine the convergence or divergence of ∞ (−1)k k=1 3k . If convergent, Þnd its value. 10. Give a rigorous proof that 0.33333. . . = 13 . This means to use inÞnite series. 11. Use inÞnite series.to Þnd the fractional value of 0.123123123123. . . 12. The Taylor series for ex , sin(x), and cos(x) about a = 0. Also know that the interval of convergence for each of these functions is−∞ < x < θ. 13. Be able to Þnd the Taylor series for ln(x) about a = 1 via diﬀerentiation. What is the interval of convergence? 14. Be able to Þnd the Taylor series for sinh(x) and the intervals of convergence? 15. Be able to Þnd the Taylor series for the interval of convergence? 1 1+x2 1 1+x at a = 0 via diﬀerentiation. What are at a = 0 using the Taylor series for 16. Know the syntax in Maple for Þnding Taylor series. 1 . 1+x What is