Stepwise problems - Thu. 11:59pm
Taylor and Maclaurin series
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01
Maclaurin series
For each of these functions, find the Maclaurin series, and the interval on which the expansion is valid.
(a)
(b) Link to originalSolution
01
(a)
(b)
Link to original
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04
Discovering the function from its Maclaurin series
For each of these series, identify the function of which it is the Maclaurin series.
(a)
(b) Link to originalSolution
02
(a) Notice the matching powers. Collect powers and then observe the geometric series pattern:
(b)
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Applications of Taylor series
03
01
Approximating
Using the series representation of
, show that: Now use the alternating series error bound to approximate
to an error within . Link to originalSolution
03
Notice that
. We have this series for
: Therefore:
Now we use the “Next Term Bound” rule. Calculate terms until we find a term less than
: So we take the following partial sum approximation:
Link to original
Regular problems - Sun. 11:59pm
Taylor and Maclaurin series
04
02
Maclaurin series
For each of these functions, find the Maclaurin series and the interval on which the expansion is valid.
(a)
(b) Link to originalSolution
04
(a)
(b)
Link to original
05
03
Taylor series of
Find the Taylor series for the function
centered at . Link to originalSolution
05
Calculate derivatives. Use
.
0 1 2 3 4 5 6 Each new derivative takes down the next power as a factor, and switches the sign. The accumulation of powers follows a factorial pattern, and these factorials cancel those added to the denominator to make
. So we have:
Alternate method:
We can derive this Taylor series using some algebraic tricks with the standard geometric series:
Link to original
06
05
Discovering the function from its Maclaurin series
For each of these series, identify the function of which it is the Maclaurin series, and evaluate the function at an appropriate choice of
to find the total sum for the series. (a)
(b) (c) Link to originalSolution
06
(a)
(b)
(c)
Link to original
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06
Summing a Maclaurin series by guessing its function
For each of these series, identify the function of which it is the Maclaurin series:
(a)
(b) Now find the total sums for these series:
(c)
(d) (Hint: for (c)-(d), do the process in (a)-(b), then evaluate the resulting function somewhere.)
Link to originalSolution
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(a)
(b)
(c)
(d)
Link to original
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07
Data of a Taylor series
Assume that
, , , and . Find the first four terms of the Taylor series of
centered at . Link to originalSolution
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Use the formula
: Therefore:
Link to original
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08
Evaluating series
Find the total sums for these series:
(a)
(b) Link to originalSolution
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(a)
Recall the series for
: This matches our series if we set
. So the total sum is:
(b)
Link to original
10
09
Large derivative at
using pattern of Maclaurin series Consider the function
. Find the value of . (Hint: find the rule for coefficients of the Maclaurin series of
and then plug in .) Link to originalSolution
10
The Coefficient-Derivative Identity says that
where is the coefficient of the power . Solve:
So, for the coefficient
: Link to original
Applications of Taylor series
11
02
Some estimates using series
Without a calculator, estimate
(angle in radians) with an error below . (Use the error bound formula for alternating series.)
Link to originalSolution
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Write the alternating series for
: This is an alternating series, so we can apply the “Next Term Bound” rule. Calculate some terms:
(Without a calculator, we can see that
. Dividing by will only decrease this value.) So we add up the prior terms:
Link to original
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03
Some estimates using series
Find an infinite series representation of
and then use your series to estimate this integral to within an error of . (Use the error bound formula for alternating series.)
Link to originalSolution
12
Write the series of the integrand:
Integrate:
Now apply the “Next Term Bound” and look for the first term below
: So we simply add the first two terms:
Link to original