Mathematics Notes | EduRev

Algebra for IIT JAM Mathematics

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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.

 Mathematics Notes | EduRev

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

 Mathematics Notes | EduRev

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

 Mathematics Notes | EduRev

Step 3. So, from the previous step we see that the radius of convergence is  Mathematics Notes | EduRev
Step 4. Now, let’s start working on the interval of convergence. Let’s break up the inequality we got in Step 2.

 Mathematics Notes | EduRev

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

 Mathematics Notes | EduRev

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. 

 Mathematics Notes | EduRev

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

 Mathematics Notes | EduRev

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

 Mathematics Notes | EduRev

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

Step 2. Therefore, we know that the interval of convergence is  Mathematics Notes | EduRev and the radius of convergence is  Mathematics Notes | EduRev

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

 Mathematics Notes | EduRev

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

 Mathematics Notes | EduRevOkay, we can see that , in this case, L=0 for every x. 

Step 2. Therefore, we know that the interval of convergence is  Mathematics Notes | EduRev and the radius of convergence is  Mathematics Notes | EduRev

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

 Mathematics Notes | EduRev

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

 Mathematics Notes | EduRev

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

 Mathematics Notes | EduRev

Step 3. So, from the previous step we see that the radius of convergence is  Mathematics Notes | EduRev

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

 Mathematics Notes | EduRev

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

 Mathematics Notes | EduRev

Now, 

 Mathematics Notes | EduRev

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.

 Mathematics Notes | EduRev

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

 Mathematics Notes | EduRev

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

 Mathematics Notes | EduRev

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

 Mathematics Notes | EduRev

Step 3. So, from the previous step we see that the radius of convergence is  Mathematics Notes | EduRev

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

 Mathematics Notes | EduRev

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

 Mathematics Notes | EduRev

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.

 Mathematics Notes | EduRev

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