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Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Document Description: Problems for Practice - 1 for Mathematics 2022 is part of Algebra for IIT JAM Mathematics preparation. The notes and questions for Problems for Practice - 1 have been prepared according to the Mathematics exam syllabus. Information about Problems for Practice - 1 covers topics like and Problems for Practice - 1 Example, for Mathematics 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Problems for Practice - 1.

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For each of the following power series determine the interval and radius of convergence.

1. For the following power series determine the interval and radius of convergence.

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Solution. Okay, let’s start off with the Ratio Test to get our hands on L.

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Step 2. So, we know that the series will converge if,

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Step 3. So, from the previous step we see that the radius of convergence is Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics
Step 4. Now, let’s start working on the interval of convergence. Let’s break up the inequality we got in Step 2.

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Step 5. To finalize the interval of convergence we need to check the end points of the inequality from Step 4.

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Now, we can do a quick Comparison Test on the first series to see that it converges and we can do a quick Alternating Series Test on the second series to see that is also converges. We’ll leave it to you to verify both of these statements.
Step 6. The interval of convergence is below and for summary purposes the radius of convergence is also shown. 

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

2. For the following power series determine the interval and radius of convergence.

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Solution. Okay, let’s start off with the Root Test to get our hands on L. 

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Okay, we can see that , in this case, L will be infinite provided Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics and so the series will diverge for Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics We also know that the power series will converge for Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics (this is the value of a for this series!).

Step 2. Therefore, we know that the interval of convergence is Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics and the radius of convergence is Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

3. For the following power series determine the interval and radius of convergence.

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Solution. Okay, let’s start off with the Ratio Test to get our hands on L. 

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - MathematicsOkay, we can see that , in this case, L=0 for every x. 

Step 2. Therefore, we know that the interval of convergence is Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics and the radius of convergence is Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

4. For the following power series determine the interval and radius of convergence. 

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Solution. Okay, let’s start off with the Ratio Test to get our hands on L. 

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Step 2. So, we know that the series will converge if,

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Step 3. So, from the previous step we see that the radius of convergence is Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Step 4. Now, let’s start working on the interval of convergence.  Let’s break up the inequality we got in Step 2.

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Step 5. To finalize the interval of convergence we need to check the end points of the inequality from Step 4. 

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Now, 

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Therefore, each of these two series diverge by the Divergence Test.

Step 6. The interval of convergence is below and for summary purposes the radius of convergence is also shown.

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

5. For the following power series determine the interval and radius of convergence.

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Solution. Okay, let’s start off with the Ratio Test to get our hands on L.

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Step 2. So, we know that the series will converge if,

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Step 3. So, from the previous step we see that the radius of convergence is Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Step 4. Now, let’s start working on the interval of convergence.  Let’s break up the inequality we got in Step 2.

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Step 5. To finalize the interval of convergence we need to check the end points of the inequality from Step 4.

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

Now, the first series is an alternating harmonic series which we know converges (or you could just do a quick Alternating Series Test to verify this) and the second series diverges by the p-series test.

Step 6. The interval of convergence is below and for summary purposes the radius of convergence is also shown.

Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics

The document Problems for Practice - 1 Notes | Study Algebra for IIT JAM Mathematics - Mathematics is a part of the Mathematics Course Algebra for IIT JAM Mathematics.
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