01
Maclaurin series
For each of these functions, find the Maclaurin series, and the interval on which the expansion is valid.
(a)
(b)
Solution
01
(a)
(b)
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02
Maclaurin series
For each of these functions, find the Maclaurin series and the interval on which the expansion is valid.
(a)
(b)
Solution
04
(a)
(b)
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03
Taylor series of
Find the Taylor series for the function
centered at .
Solution
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:
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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)
Solution
02
(a) Notice the matching powers. Collect powers and then observe the geometric series pattern:
(b)
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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)
Solution
06
(a)
(b)
(c)
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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.)
Solution
07
(a)
(b)
(c)
(d)
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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 .
Solution
08
Use the formula
: Therefore:
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08
Evaluating series
Find the total sums for these series:
(a)
(b)
Solution
09
(a)
Recall the series for
: This matches our series if we set
. So the total sum is:
(b)
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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 .)
Solution
10
The Coefficient-Derivative Identity says that
where is the coefficient of the power . Solve:
So, for the coefficient
: Link to original