Step by step pls

MATH 2414 - Review Exam 3 Please use this review (along with your notes and homework) to prepare for the exam. Determine if the following sequences whose nth terms are given below converge or diverge. Find the limit of each convergent sequence. 1. a,, = sin nit 2 2 . a 3. a =- (n +1)! n! Determine the convergence or divergence of the following problems (check for conditional convergence on alternating series, if needed). Identify the test used and JUSTIFY YOUR ANSWERS 4. (-1)" 5. 6 . 5 "=in+3 n=i n' +10 moe 1 7. 3. 1=2 n (Inn) E- 1 1 71=1 6" +2 9 . S- "=0 Vn2 +6 COS (IT) 10. 11. (2n)! -3n no n'+1 m=i n8 12. Z 2n + 1, 13. S- 4 14. n 15. _ 2n + 5 n=1 no 7n' + 6n +1 Find the interval of convergence of the power series. (Be sure to include a check for convergence at the end points of the interval.) 16. 1=1 (3n)! 17. (2n)!(x+4)" 18. (-1)" (x-3)" 1=0 5" 1=0 (n +1)7" Find a power series for the function centered at c and determine the interval of convergence. 19. f (x ) = - 3 , C=-3 2 5 -x 20. f (x) = - , C=0 21. f (x) = 7 2+ x2 2x -5 , C=6 22. Find the nth Taylor polynomial for the function centered at c: f (x) = sinx, n =5, c= 3 TT 4 Use the definition of the Taylor Series to find the Taylor series, centered at c for the function. 23. f (x) = =, c=1 24. f ( x) = 3" , c= 0 Use the binomial series to find the Maclaurin series for the function: 25. f (x) =- (1-x) 26. f (x ) =- V4+ x 2 Use the table of power series for elementary functions to find the Maclaurin series for the following functions. 27. Cos (3x) 28. In (1+x ) 29. xe 2 x 2 Use a power series to approximate the value of the definite integral with an error of less than 0.0001. 30. ex' dx 31. [xin(x+1) dx