Stepwise problems - Thu. 11:59pm
Power series as functions
01
Modifying geometric power series
Consider the geometric power series
for . For this problem, you should modify the series for
. (a) Write
as a power series centered at and determine its radius of convergence. (b) Write
as a power series centered at and determine its radius of convergence.
Tayler and Maclaurin series
02
Maclaurin series I
For each of these functions, find the Maclaurin series.
(a)
(b)
Applications of Taylor series
03
Approximating
Using the series representation of
, show that: Now use the alternating series error bound to approximate
to an error within .
Regular problems - Sat. 11:59pm
Power series as functions
04
Power series of a derivative
Suppose that a function
has power series given by: The radius of convergence of this series is
. What is the power series of
and what is its radius of convergence?
05
Finding a power series
Find a power series representation for these functions:
(a)
(b)
06
Modifying and integrating a power series
(a) Modify the power series
for to obtain the power series for . (b) Now integrate this series to find the power series for
.
Tayler and Maclaurin series
07
Maclaurin series II
For each of these functions, find the Maclaurin series.
(a)
(b)
08
Taylor series of
Find the Taylor series for the function
centered at .
09
Discovering the function from its Maclaurin series I
For each of these series, identify the function of which it is the Maclaurin series.
(a)
(b)
10
Discovering the function from its Maclaurin series II
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)
11
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.)
12
Data of a Taylor series
Assume that
, , , and . Find the first four terms of the Taylor series of
centered at .
13
Evaluating series
Find the total sums for these series:
(a)
(b)
14
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 .)
Applications of Taylor series
15
Some estimates using series
For each of these estimates, use the error bound formula for alternating series.
Without a calculator, estimate
(angle in radians) with an error below .
16
Some estimates using series
For each of these estimates, use the error bound formula for alternating series.
Find an infinite series representation of
and then use your series to estimate this integral to within an error of .