On page 85, a power series is given. Use the root test to prove its convergence or divergence and determine its radius of convergence. Instructions: Solve step-by-step. Apply the root test rigorously, and show all intermediate calculations. Avoid irrelevant details. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440AZF/view?usp=sharing]

Refer to page 86 for a problem involving solving Legendre's differential equation. Instructions: Solve using power series or standard solutions. Clearly justify every step and avoid unnecessary explanations. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qo Hazb9tC440 AZF/view?usp=sharing] Refer to page 80 for a proof of convergence for a given series using the ratio test. Instructions: Clearly apply the ratio test. Show all steps and provide justification for convergence or divergence. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]

Determine whether the series is absolutely convergent, conditional convergent or divergent. Make sure to show detail step by step solutions. Make sure to name the test that you are using to do this problem, example: integral test, comparison test, alternating series test, root test, etc. n=1 (−1)n-³ √n

Refer to page 80 for a proof of convergence for a given series using the ratio test. Instructions: Clearly apply the ratio test. Show all steps and provide justification for convergence or divergence. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440 AZF/view?usp=sharing] Refer to page 94 for a problem requiring the numerical solution of an ODE using the Runge- Kutta method. Instructions: Solve step-by-step, showing iterations, step sizes, and calculations clearly. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qo Hazb9tC440AZF/view?usp=sharing]

Determine whether the series is avsolutely convergent conditionally convergent or divergent. If posible, please check for absolute convergence first.

Use the limit comparison test or comparison test to determine if the series converges or diverges. En=0 4n+I a. Demonstrate a limit or inequality that compares the given pattern to another series. Make sure to state what that series is that you are using for comparison. b. Address the convergence of divergence of your comparison series and how it relates to the convergence or divergence of your given series.

You add a finite number of terms to a convergent series. Will the new series still converge? Explain your reasoning.

Give an example of a series that converges by the integral test.

can you please answer both questions. Decide whether those two series will converge. Thank you

Determine whether the series is convergent or divergent.