Posted: June 16th, 2023

Series. Determine if the series converges or diverges.

Math 1452-01011/28/2015Written HW #2 (Extra Credit)Due Date: Wednesday, December 9th at 4:30 PM (i.e. due at the final exam)Directions: Please organize your work, write legibly, and solve the problems carefully. Include each major stepand justify your work as much as you reasonably can. This assignment is worth a maximum of 50 points to thewritten homework category.1. Series. Determine if the series converges or diverges. If the series converges, determine if it convergesabsolutely or conditionally. If the series diverges, choose and justify the most appropriate answer from thefollowing three choices: ∞, −∞, or . (5 points each)(a)(b)(c)∞∞∞(−1) +1sin()sin2 ()∑∑ 1.001∑ln⁡ )(k!=1=1=02. Power Series. Determine the convergence set and specify any points where the convergence isconditional. (5 points each)(a)(b)(c)∞∞∞∑ 2∑ !∑ln()=1=0=13. Taylor/Maclaurin Series. (5 points each)(a) Write the Maclaurin series for () = cos2() using sigma notation. (Hint: Use the identity cos2 () =1+cos(2))2(b) Write the Taylor series for () = ( − 1) at = 1 using sigma notation. (Hint: First find the Taylorseries for at = 1)4. Vectors. (5 points each)(a) True/False. For vectors ⃗, in ℝ3 , we have ⃗ ∙ = 0 if and only if ⃗ = 0 or = 0, where the dot meansdot product. Explain your answer. (See section 9.3 for the definition of the dot product.)(b) Give an example which demonstrates that the cross product of vectors in ℝ3 is not associative. (Seesection 9.4 for the definition of the cross product. We will encounter this on the last day of class.)

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