Mathematics Notes | EduRev

Algebra for IIT JAM Mathematics

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The document Mathematics Notes | EduRev is a part of the Mathematics Course Algebra for IIT JAM Mathematics.
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For problems 1 – 3 write the given function as a power series and give the interval of convergence.

1. Write the following function as a power series and give the interval of convergence. 

 Mathematics Notes | EduRev

Solution. First, in order to use the formula from this section we know that we need the numerator to be a one. That is easy enough to “fix” up as follows,

 Mathematics Notes | EduRev

Step 2. Next, we know we need the denominator to be in the form 1−p and again that is easy enough, in this case, to rewrite the denominator to get the following form of the function, 

 Mathematics Notes | EduRev

Step 3. At this point we can use the formula from the notes to write this as a power series.  Doing this gives, 

 Mathematics Notes | EduRev

Step 4. Now, recall the basic “rules” for the form of the series answer. We don’t want anything out in front of the series and we want a single x with a single exponent on it.
These are easy enough rules to take care of. All we need to do is move whatever is in front of the series to the inside of the series and use basic exponent rules to take care of the x “rule”. Doing all this gives,

 Mathematics Notes | EduRev

Step 5. To get the interval of convergence all we need to do is do a little work on the “provided” portion of the result from the last step to get,

 Mathematics Notes | EduRev

Note that we don’t need to check the endpoints of this interval since we already know that we only get convergence with the strict inequalities and we will get divergence for everything else.

Step 6. The answers for this problem are then,

 Mathematics Notes | EduRev

2. Write the following function as a power series and give the interval of convergence. 

 Mathematics Notes | EduRev

Solution. First, in order to use the formula from this section we know that we need the numerator to be a one.  That is easy enough to “fix” up as follows,

 Mathematics Notes | EduRev

Step 2. Next, we know we need the denominator to be in the form 1−p and again that is easy enough, in this case, to rewrite the denominator by factoring a 3 out of the denominator as follows,

 Mathematics Notes | EduRev

Step 3. At this point we can use the formula from the notes to write this as a power series.  Doing this gives,

 Mathematics Notes | EduRev

Step 4. Now, recall the basic “rules” for the form of the series answer.  We don’t want anything out in front of the series and we want a single x
with a single exponent on it.
These are easy enough rules to take care of.  All we need to do is move whatever is in front of the series to the inside of the series and use basic exponent rules to take care of the x
“rule”.  Doing all this gives,

 Mathematics Notes | EduRev

Step 5. To get the interval of convergence all we need to do is do a little work on the “provided” portion of the result from the last step to get,

 Mathematics Notes | EduRev

Note that we don’t need to check the endpoints of this interval since we already know that we only get convergence with the strict inequalities and we will get divergence for everything else.
Step 6. The answers for this problem are then,

 Mathematics Notes | EduRev

3. Write the following function as a power series and give the interval of convergence. 

 Mathematics Notes | EduRev

Solution. First, in order to use the formula from this section we know that we need the numerator to be a one.  That is easy enough to “fix” up as follows,

 Mathematics Notes | EduRev

Step 2. Next, we know we need the denominator to be in the form 1−p and again that is easy enough, in this case, to rewrite the denominator by factoring a 5 out of the denominator as follows, 

 Mathematics Notes | EduRev

Step 3. At this point we can use the formula from the notes to write this as a power series.  Doing this gives, 

 Mathematics Notes | EduRev

Step 4. Now, recall the basic “rules” for the form of the series answer.  We don’t want anything out in front of the series and we want a single x with a single exponent on it.
These are easy enough rules to take care of.  All we need to do is move whatever is in front of the series to the inside of the series and use basic exponent rules to take care of the x “rule”.  Doing all this gives,

 Mathematics Notes | EduRev

Step 5. To get the interval of convergence all we need to do is do a little work on the “provided” portion of the result from the last step to get,

 Mathematics Notes | EduRev

Note that we don’t need to check the endpoints of this interval since we already know that we only get convergence with the strict inequalities and we will get divergence for everything else.

Step 6. The answers for this problem are then, 

 Mathematics Notes | EduRev

4. Give a power series representation for the derivative of the following function.

 Mathematics Notes | EduRev

Solution. First let’s notice that we can quickly find a power series representation for this function. Here is that work.

 Mathematics Notes | EduRev

Step 2. Now, we know how to differentiate power series and we know that the derivative of the power series representation of a function is the power series representation of the derivative of the function.
Therefore,

 Mathematics Notes | EduRev

Remember that to differentiate a power series all we need to do is differentiate the term of the power series with respect to x. 

5. Give a power series representation for the integral of the following function.

 Mathematics Notes | EduRev

Solution. First let’s notice that we can quickly find a power series representation for this function. Here is that work. 

 Mathematics Notes | EduRev

Step 2. Now, we know how to integrate power series and we know that the integral of the power series representation of a function is the power series representation of the integral of the function.

 Mathematics Notes | EduRev

Remember that to integrate a power series all we need to do is integrate the term of the power series and we can’t forget to add on the “+c” since we’re doing an indefinite integral.

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