# Calculus 2 : Maclaurin Series

## Example Questions

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### Example Question #1 : Maclaurin Series

Suppose that . Calculate

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?

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?

The infinite series diverges.

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.

The series does not converge.

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:

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.

Divergent.

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.

Infinite series does not converge.

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 #8 : Maclaurin Series

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

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 #9 : Maclaurin Series

Find the Maclaurin series for the function:

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 #10 : Maclaurin Series

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