Sequences
01
Link to originalL’Hopital practice - converting indeterminate form
By imitating the technique of the L’Hopital’s Rule example, find the limit of the sequence:
06
Link to originalLimits and convergence
For each sequence, either write the limit value (if it converges), or write ‘diverges’.
(a)
(b) (c) (d) (e)
(f) (g) (h)
Series basics
02
Link to originalGeometric series
Compute the following summation values using the sum formula for geometric series.
(a)
(b)
03
Link to originalGeometric series
Compute the following summation values using the sum formula for geometric series.
(a)
(b)
Positive series
05
Link to originalIntegral Test, Direct Comparison Test, Limit Comparison Test
Determine whether the series converges by checking applicability and then applying the designated convergence test.
(a) Integral Test:
(b) Direct Comparison Test:
(c) Limit Comparison Test:
03
Link to originalLimit Comparison Test (LCT)
Use the Limit Comparison Test to determine whether the series converges:
Show your work. You must check that the test is applicable.
Alternating series
01
Link to originalAbsolute and conditional convergence
Determine whether the series are absolutely convergent, conditionally convergent, or divergent.
Show your work. You must check applicability of tests.
(a)
(b)
03
Link to originalAlternating series: error estimation
Find the approximate value of
such that the error satisfies . How many terms do you really need?
Ratio test and Root test
02
Link to originalRatio and root tests
Apply the ratio test or the root test to determine whether each of the following series is absolutely convergent, conditionally convergent, or divergent.
(a)
(b) (c)
Power series: Radius and Interval
02
Link to originalPower series - radius and interval
Find the radius and interval of convergence for these power series:
(a)
(b) (c)
Series tests: strategy tips
02
Link to originalVarious limits, Part II
Find the limits. You may use
or or as appropriate. Braces indicate sequences.
- C = Convergent
- AC = Absolutely Convergent
- CC = Conditionally Convergent
- D = Divergent
C or D
C or D
AC, CC, or D
AC, CC, or D
Power series as functions
01
Link to originalModifying geometric power series
Consider the geometric power series
for . For this problem, you should modify the series for
. (a) Write
as a power series and determine its interval of convergence. (b) Write
as a power series and determine its interval of convergence.
03
Link to originalFinding a power series
Find a power series representation for these functions:
(a)
(b)
Taylor and Maclaurin series
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)
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)
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.)
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 .)
Applications of Taylor series
01
Link to originalApproximating
Using the series representation of
, show that: Now use the alternating series error bound to approximate
to an error within .
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.)