Stepwise problems - Thu. 11:59pm
Sequences
01
01
L’Hopital practice - converting indeterminate form
By imitating the technique of the L’Hopital’s Rule example, find the limit of the sequence:
Solution
01
(1) Indeterminate form:
(2) L’Hopital:
Convert:
Change to
and apply L’Hopital:
(3) Take limit:
Therefore
Link to originalas .
02
05
Limits and convergence
For each sequence, either write the limit value (if it converges), or write ‘diverges’.
(a)
(b) (c) (d)
(e) Solution
02
(a)
(b) diverges (c) (d)
Observe that
as , but for each , the value is below , in the domain of , which is continuous for . (e)
Link to original
Series
03
01
General term of a series
Write this series in summation notation:
(Hint: Find a formula for the general term
.) Link to originalSolution
Regular problems - Next Wed. 11:59pm
Sequences
04
03
Computing the terms of a sequence
Calculate the first four terms of each sequence from the given general term, starting at
: (a)
(b) (c) (d) (e) (f) Link to originalSolution
04
(a)
(b)
(c)
(d)
(e)
(f)
Link to original
05
04
General term of a sequence
Find a formula for the general term (the
term) of each sequence: (a)
(b) (c) Link to originalSolution
05
(a)
(b)
(c)
Link to original
06
02
Squeeze theorem
Determine whether the sequence converges, and if it does, find its limit:
(a)
(b) (Hint for (b): Verify that
.) Link to originalSolution
06
(a)
(1) Set up squeeze relations:
(2) Apply theorem:
We have:
Therefore:
We conclude that
converges.
(b)
(1) Generate squeeze inequalities:
Observe:
Rewrite RHS:
Raise all terms to
:
(2) Apply squeeze theorem:
Therefore:
Conclude that:
Link to original
Series
07
05
Series from its partial sums
Suppose we know that the partial sums
of a series are given by the formula . (a) Compute
. (b) Find a formula for the general term
. (c) Find the sum
. Link to originalSolution
07
(a)
(b)
When
, this formula is undefined, because is undefined. But we know that:
(c)
Simply take the limit of
as : Link to original
08
06
Geometric series - partial sums and total sum
Consider the series:
(a) Compute a formula for the
partial sum . (You may apply the known formula or derive it again in this case using the “shift method.”) (b) By taking the limit of this formula as
, find the value of the series. (c) Find the same value of the series by computing
and and plugging into . Link to originalSolution
08
(a)
(1) Recall geometric partial sum formula:
This one may be easiest to recall:
(Note here
is the first term in the summation so it appears in the formula.)
(2) Identify ingredients in partial sum formula:
Rewrite summand to determine
and : We see that
and .
(b) Take limit:
Note A: The second term goes to zero:
.
(c)
Link to original