By Nancy Novak
Final Russian has every thing you want to research Russian from scratch or to restore the Russian that you just discovered years in the past. This direction combines dialog and tradition in an easy-to-follow, relaxing, and potent layout. Its definitely the right technique to study Russian for college, commute, paintings, or own enrichment.
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Extra info for Ultimate Russian Beginner-Intermediate
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3n ϩ 1 (a) Determine whether ͕a n ͖ is convergent. (b) Determine whether ϱn1 a n is convergent. ͚ n1 ϱ 9. Let a n ■ ϱ ͚ n1 ■ 1 ͚ (s2 ) n0 k2 2 k Ϫ1 k2 ■ ͑Ϫ6͒ nϪ1 5 nϪ1 ϱ 18. ϱ 1 n͑n Ϫ 1͒ ͚ 8. ■ n1 1 2n ͚ n1 n1 ϱ ■ ͚ 17–26 Determine whether the series is convergent or divergent. If it is convergent, find its sum. 6͒ 6. 16. ͚ n0 ϱ ͚ tan n n1 n 3 nϩ1 ϱ ; 3–8 12 ͑Ϫ5͒n 14. 2 nϪ1 3 n1 2. Explain what it means to say that ϱn1 a n 5. ϱ ϱ ͚ 5( ) ■ 31–34 ϱ 2 n Ϫ1 28. 3 n͑n ϩ 3͒ 30. 2 ■ n1 ■ n ͚ ln n ϩ 1 n1 ■ ■ ■ ■ ■ ■ ■ ■ ■ Express the number as a ratio of integers.
1 ϩ ϱ i 12. i2 ϱ bn is known to be convergent. 13. (a) If a n Ͼ bn for all n, what can you say about a n? Why? (b) If a n Ͻ bn for all n, what can you say about a n? Why? 15. ϱ 17. ͚ n1 ϱ 19. 5. It is important to distinguish between ͚ n1 ϱ ϱ ͚ n1 ͚ n2 (a) If a n Ͼ bn for all n, what can you say about an ? Why? (b) If a n Ͻ bn for all n, what can you say about an ? Why? ͚ 21. bn ͚ n1 n1 ϱ What name is given to the first series? To the second? For what values of b does the first series converge?
8. ͕5, 1, 5, 1, 5, 1, . ͖ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. ) 5–8 {1, Ϫ 23 , 49 , Ϫ 278 , . } ■ 1ϩ ; 29–34 this sequence appear to have a limit? If so, find it. If not, explain why. 5. ͩ ͪ 2 n n sin 2n 1 ϩ sn (b) What is a divergent sequence? Give two examples. 3. List the first six terms of the sequence defined by 22.