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Example Questions
Example Question #1 : Maclaurin Series
Suppose that
. Calculate .
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 #1 : Maclaurin Series
What is the value of the following infinite series?
We can recognize this series as
since the power series is
with the value
plugged into since.
So then we have
.
Example Question #1 : Maclaurin Series
What is the value of the following infinite series?
The infinite series diverges.
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.
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:
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.
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.
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 #1 : Maclaurin Series
Write out the first two terms of the Maclaurin series of the following function:
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 #3091 : Calculus Ii
Find the Maclaurin series for the function:
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:
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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