Stepwise problems - Thu. 11:59pm
Taylor and Maclaurin series
01
01
Link to originalMaclaurin series
For each of these functions, find the Maclaurin series, and the interval on which the expansion is valid.
(a)
(b)
Solution
02
04
Link to originalDiscovering the function from its Maclaurin series
For each of these series, identify the function of which it is the Maclaurin series.
(a)
(b)
Solution
Applications of Taylor series
03
01
Link to originalApproximating
Using the series representation of
, show that: Now use the alternating series error bound to approximate
to an error within .
Solution
Regular problems - Sun. 11:59pm
Taylor and Maclaurin series
04
02
Link to originalMaclaurin series
For each of these functions, find the Maclaurin series and the interval on which the expansion is valid.
(a)
(b)
Solution
05
03
Link to originalTaylor series of
Find the Taylor series for the function
centered at .
Solution
06
05
Link to originalDiscovering 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
07
06
Link to originalSumming 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
08
07
Link to originalData of a Taylor series
Assume that
, , , and . Find the first four terms of the Taylor series of
centered at .
Solution
09
08
Link to originalEvaluating series
Find the total sums for these series:
(a)
(b)
Solution
10
09
Link to originalLarge 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
Applications of Taylor series
11
02
Link to originalSome estimates using series
Without a calculator, estimate
(angle in radians) with an error below . (Use the error bound formula for alternating series.)
Solution
12
03
Link to originalSome 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.)
Solution