Sequences
01
Link to originalL’Hopital practice - converting indeterminate form
By imitating the technique in from the L’Hopital’s Rule example, find the limit of the sequence:
Series basics
04
Link to originalGeometric series
Compute the following summation values using the sum formula for geometric series.
(a)
(b) (c) (d)
07
Link to originalLimits and convergence
For each sequence, either write the limit value (if it converges), or write ‘diverges’.
(a)
(b) (c) (d) (d)
(e) (f) (g) (h)
(i) (j) (k) (l)
(m) (n) (o) (p)
(q) (r) (s) (t)
Positive series
06
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
04
Link to originalAbsolute and conditional convergence
Apply the Alternating Series Test (AST) to determine whether the series are absolutely convergent, conditionally convergent, or divergent.
Show your work. You must check that the test is applicable.
(a)
(b)
09
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
03
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
06
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
05
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 centered at and determine its radius of convergence. (b) Write
as a power series centered at and determine its radius of convergence.
05
Link to originalFinding a power series
Find a power series representation for these functions:
(a)
(b)
Taylor and Maclaurin series
02
Link to originalMaclaurin series I
For each of these functions, find the Maclaurin series.
(a)
(b)
10
Link to originalDiscovering 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
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.)
14
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
03
Link to originalApproximating
Using the series representation of
, show that: Now use the alternating series error bound to approximate
to an error within .
16
Link to originalSome 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 .