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Problems for Practice - 1 | Algebra - Mathematics PDF Download

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 | Algebra - Mathematics

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

Problems for Practice - 1 | Algebra - Mathematics

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

Problems for Practice - 1 | Algebra - Mathematics

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

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

Problems for Practice - 1 | Algebra - Mathematics

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

Problems for Practice - 1 | Algebra - Mathematics

Okay, we can see that , in this case, L will be infinite provided Problems for Practice - 1 | Algebra - Mathematics and so the series will diverge for Problems for Practice - 1 | Algebra - Mathematics We also know that the power series will converge for Problems for Practice - 1 | Algebra - 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 | Algebra - Mathematics and the radius of convergence is Problems for Practice - 1 | Algebra - Mathematics

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

Problems for Practice - 1 | Algebra - Mathematics

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

Problems for Practice - 1 | Algebra - 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 | Algebra - Mathematics and the radius of convergence is Problems for Practice - 1 | Algebra - Mathematics

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

Problems for Practice - 1 | Algebra - Mathematics

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

Problems for Practice - 1 | Algebra - Mathematics

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

Problems for Practice - 1 | Algebra - Mathematics

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

Now, 

Problems for Practice - 1 | Algebra - 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 | Algebra - Mathematics

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

Problems for Practice - 1 | Algebra - Mathematics

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

Problems for Practice - 1 | Algebra - Mathematics

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

Problems for Practice - 1 | Algebra - Mathematics

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

The document Problems for Practice - 1 | Algebra - Mathematics is a part of the Mathematics Course Algebra.
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FAQs on Problems for Practice - 1 - Algebra - Mathematics

1. What are the different branches of mathematics?
Ans. Mathematics is divided into various branches such as algebra, geometry, calculus, statistics, and number theory. Each branch focuses on different aspects of mathematical concepts and applications.
2. How can I improve my problem-solving skills in mathematics?
Ans. To enhance problem-solving skills in mathematics, it is important to practice regularly, understand the underlying concepts, break down complex problems into smaller parts, seek help when needed, and learn from mistakes. Additionally, engaging in problem-solving activities and participating in math competitions can also contribute to skill development.
3. What is the importance of mathematics in everyday life?
Ans. Mathematics plays a crucial role in our daily lives. It helps us in managing finances, making informed decisions, measuring quantities, understanding patterns, solving real-life problems, and developing critical thinking skills. From calculating grocery bills to planning budgets, mathematics is essential in numerous aspects of our routine activities.
4. How can I overcome math anxiety?
Ans. Math anxiety is a common issue faced by many individuals. To overcome it, try to identify the root causes of anxiety, practice relaxation techniques, break down complex problems into simpler steps, build a positive mindset towards mathematics, seek support from teachers or tutors, and celebrate small victories during the learning process. Consistent practice and exposure to various math problems can also help in reducing math anxiety.
5. How can I prepare effectively for a mathematics exam?
Ans. Effective preparation for a mathematics exam involves creating a study schedule, reviewing the syllabus and important topics, practicing previous year question papers, solving sample papers, seeking clarification on doubts, understanding the marking scheme, and revising regularly. It is also beneficial to form study groups, teach others, and engage in discussions to reinforce understanding and gain different perspectives on problem-solving techniques.
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