SalasSv_11_01_ex - EXERCISES 11.1 Evaluate the given...

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Unformatted text preview: EXERCISES 11.1 Evaluate the given expression. 1. 2 k = (3 k + 1). 2. 4 k = 1 (3 k 1). 3. 3 k = 2 k . 4. 3 k = ( 1) k 2 k + 1 . 5. 5 k = 3 ( 1) k k ! . 6. 4 k = 2 1 3 k 1 . Express in sigma notation. 7. 1 + 3 + 5 + 7 + + 21. 8. 1 3 + 5 7 + 19. 9. 1 2 + 2 3 + 3 4 + + 35 36. 10. The lower sum m 1 x 1 + m 2 x 2 + + m n x n . 11. The upper sum M 1 x 1 + M 2 x 2 + + M n x n . 12. The Riemann sum f ( x 1 ) x 1 + f ( x 2 ) x 2 ++ f ( x n ) x n . Write the given sums as 10 k = 3 a k and 7 i = a i + 3 . 13. 1 2 3 + 1 2 4 + + 1 2 10 . 14. 3 3 3 ! + 4 4 4 ! + + 10 10 10 ! . 15. 3 4 4 5 + 10 11 . 16. 1 3 + 1 5 + 1 7 + + 1 17 . Verify by a change of indices that the two sums are identical. 17. 10 k = 2 k k 2 + 1, ; 7 n = 1 n + 3 n 2 + 6 n + 10 . 18. 12 n = 2 ( 1) n n 1. ; 11 k = 1 ( 1) k + 1 k . 19. 25 k = 4 1 k 2 9 ; 28 n = 7 1 n 2 6 n . 20. 15 k = 3 2 k k ! ; (81) 13 n = 2 3 2 n ( n + 2) ! . The following formulas can be verified by mathematical induction: n k = 1 k = n ( n + 1) 2 , n k = 1 (2 k 1) = n 2 , n k = 1 k 2 = n ( n + 1)(2 n + 1) 6 , n k = 1 k 3 = n k = 1 k 2 . Use these formulas to evaluate the sums. 21. 10 k = 1 (2 k + 3). 22. 10 k = 1 (2 k 2 + 3 k ). 23. 8 k = 1 (2 k 1) 2 . 24. n k = 1 k ( k 2 5). Find the sum of the series. 25. k = 1 1 2 k ( k + 1) . 26. k = 3 1 k 2 k . 27. k = 1 1 k ( k + 3) . 28....
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This note was uploaded on 10/12/2010 for the course MATH 12345 taught by Professor Smith during the Spring '10 term at University of Houston - Downtown.

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SalasSv_11_01_ex - EXERCISES 11.1 Evaluate the given...

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