Calculus 2 : Maclaurin Series

Study concepts, example questions & explanations for Calculus 2

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Example Questions

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Example Question #1 : Maclaurin Series

Suppose that . Calculate 

Possible Answers:

Correct answer:

Explanation:

Let's find the power series of  centered at  to find . We have

This series is much easier to differentiate than the expression . We must look at term , which is the only constant term left after differentiating 48 times. This is the only important term, because when we plug in , all of the non-constant terms are zero. So we must have

Example Question #2 : Maclaurin Series

What is the value of the following infinite series?

Possible Answers:

Correct answer:

Explanation:

We can recognize this series as  since the power series is

with the value  plugged into  since

.

So then we have

.

Example Question #3 : Maclaurin Series

What is the value of the following infinite series?

Possible Answers:

The infinite series diverges.

Correct answer:

Explanation:

The infinite series can be computed easily by splitting up the two components of the numerator:

Now we recall the MacLaurin series for the exponential function , which is 

which converges for all . We can see that the two infinite series are  with , respectively. So we have

Example Question #4 : Maclaurin Series

Find the value of the infinite series.

Possible Answers:

The series does not converge.

Correct answer:

Explanation:

We can evaluate the series

by recognizing it as a power series of a known function with a value plugged in for . In particular, it looks similar to :

After manipulating the series, we get

.

Now it suffices to evalute , which is .

So the infinite series has value

.

Example Question #5 : Maclaurin Series

Find the value of the following infinite series:

Possible Answers:

Correct answer:

Explanation:

After doing the following manipulation:

We can see that this is the power series 

 with  plugged in.

So we have

Example Question #6 : Maclaurin Series

Find the value of the following series.

Possible Answers:

Divergent.

Correct answer:

Explanation:

We can split up the sum to get 

.

We know that the power series for  is 

and that each sum, 

 

and

 

are simply  with  plugged in, respectively.

Thus, 

.

Example Question #7 : Maclaurin Series

Find the value of the infinite series.

Possible Answers:

Infinite series does not converge.

Correct answer:

Explanation:

The series 

 looks similar to the series for , which is 

but the series we want to simplify starts at , so we can fix this by adding a  and subtracting a , to leave the value unchanged, i.e., 

.

So now we have  with , which gives us .

So then we have:

Example Question #61 : Taylor And Maclaurin Series

Write out the first two terms of the Maclaurin series of the following function:

Possible Answers:

Correct answer:

Explanation:

The Maclaurin series of a function is simply the Taylor series of a function, but about x=0 (so a=0 in the formula):

To write out the first two terms (n=0 and n=1), we must find the first derivative of the function (because the zeroth derivative is the function itself):

The derivative was found using the following rule:

Next, use the general form, plugging in n=0 for the first term and n=1 for the second term:

 

Example Question #62 : Taylor And Maclaurin Series

Find the Maclaurin series for the function:  

Possible Answers:

Correct answer:

Explanation:

Write Maclaurin series generated by a function f.  The Maclaurin series is centered at  for the Taylor series.

Evaluate the function and the derivatives of  at .

Substitute the values into the power series.  The series pattern can be seen as alternating and increasing order.

Example Question #63 : Taylor And Maclaurin Series

Find the first three terms of the Maclaurin series for the following function:

Possible Answers:

Correct answer:

Explanation:

The Maclaurin series of a function is simply the Taylor series for the function about a=0:

First, we can find the zeroth, first, and second derivatives of the function (n=0, 1, and 2 are the first three terms). 

Plugging these values into the formula we get the following.

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