By Arnold Ostebee, Paul Zorn
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E) Show that ar k . (a) Find a formula for Sn when r = 1. Explain why the formula for Sn (Equation 5) does not hold in this case. (b) Show that Sn − r Sn = (1 − r )Sn for any r . (c) Use part (b) to show that Equation 5 holds if r = 1. 10. Use Equation 5 to evaluate 3 + 6 + 12 + 24 + 48 + 96 + · · · + 3072. ] In Exercises 11–18, find the limit of the series. 1 1 1 1 1 + + + + · · · + i+4 + · · · 11. 16 32 64 128 2 1 1 1 1 1 + +···+ n +··· 12. 2 − 5 + 9 + + + 3 9 27 81 3 4. Consider the series aj = j=0 ∞ j=0 ∞ e−n 14.
Y y = a(x) The shaded part is the tail Rn. an an + 1 an + 1 n FIG URE 2 ∞ an + 2 a n+ 3 n+1 n+2 n+3 ∞ ak ≤ Comparing tails: Why k = n+1 x a(x) dx n Comparing areas in Figure 2 shows that, for all n, the tail Rn satisfies Rn = ∞ k=n+1 ∞ ak ≤ a(x) d x. n L e s s o n s f r o m t h e p i c t u r e s Figures 1 and 2 show a particular function a(x), but the conclusions we drew hold for any function a(x) that is both positive and decreasing. It is time to collect our observations in a theorem. 3 Testing for Convergence; Estimating Limits Suppose that, for all x ≥ 1, the function a(x) is continuous, positive, and decreasing.
Consider the series ak = k=1 e π k 2 3 i 3j + 4j 5j 1 diverges. 2 + sin k (−1)k converge? Justify your answer. 20. Does k=0 In Exercises 21–24, find an expression for the partial sum Sn of the series. Use this expression to determine whether the series converges and, if so, to find its limit. ∞ 21. 2 Infinite Series, Convergence, and Divergence ∞ 22. j=1 ∞ 24. ∞ j ( j + 1)! ln 1 + j=1 m=1 ∞ 1 j [HINT: (c) Evaluate lim ak . 1 1 √ −√ m m +2 23. k→∞ (d) Show that ak > 0 for all k ≥ 1. ] n 1 1 and let Sn = .