Math 152 - Spring 2008
Challenge for Exam 3 (this is an addition to the homework, not a substitute!)
1. Consider the sequence an =
e
a) Show that an+1 = an n+1
en
n! .
(1)
b) Use the previous identity to show that the sequence is decreasing for n > N (find this N ).
c) Is the sequence bounded from below?
d) Does this sequence have a limit? If so, show how one can find it using identity (1).
√
2. Let c be a positive number. Consider
the sequence bn , defined as follows: b1 = c, b2 =
q
p
p
√
√
√
√
√
c + b1 = c + c, b3 = c + b2 = c + c + c, ... or bn+1 = c + bn (2)
a) Is this sequence monotonic?
b) Show (using method of mathematical induction) that bn <
√
c+1
c) Does this sequence have a limit? If so, show how one can find it using identity (2).
3. a) Show that the sequence with the general term an = 1 +
(Hint 1. Show that is it monotonic and bounded. Hint 2.
fraction decomposition.
b) Show that the sequence with the general term an = 1 +
1
22
1
n2
+
1
32
<
1
n(n−1) ,
1
22
+
1
33
+ ... +
1
n2
converges.
and then try partial
+ ... +
1
nn
converges.
c) What application do your results in a), b) have for certain series? Write the series. Make
a conclusion about their convergence and their sums, if possible.
4. Find the infinite series and its sum if the sequence Sn of its partial sums is given as a)
2n −1
Sn = n+1
n , b) Sn = 2n , n = 1, 2, ...
5. Find the sum
∞
P
n=1
1
4n2 −1
6. Test for convergence:
n(n+1)
∞ P
n
a)
n+1
b)
n=1
∞
P
n=1
1
n
ln (1 + n1 )
c) Determine the values of real number a for which the series
∞ P
n=1
an
n+1
n
are absolutely
convergent, conditionally convergent, or divergent.
7. If f (x) = cos(x3 ), find f (6) (0) without differentiating the function.
√
8. a) Construct the Taylor series for eiθ where i = −1 by substituting x = iθ instead of x into
the expansion of ex . Using the fact that i0 = 1, i1 = i, i2 = −1, i3 = −i, i4 = 1, etc., rewrite
the series so that only the first power of i is involved.
b) Separate real and imaginary parts in the series for eiθ and verify the Euler’s formula
eiθ = cos θ + isin θ
9. Find
∞ (ln 3)n
P
n!
1
10. For a sequence of numbers a1 , a2 , a3 , ..., the infinite product
the limit of the partial products
a) Find
∞
Q
n=2
∞
Q
n
Q
∞
Q
an = a1 ×a2 ×... is defined as
n=1
ak as n → ∞ (if the limit exists, the product converges).
n=1
(1 −
1
)
k2
Q
n
k
(1 + x2 ), |x| < 1 (so you need to find the limit of Pn = nn=0 (1 + x2 ))
n=0
Q Q
Q
Idea 1: try to find a pattern as you compute 1 , 2 , ..., n , may be you will recognize
something
familiar. Idea 2. multiply each term of the product by its conjugate and then find
Q
n and its limit.
b) Find
11. Find the mistake made in the following reasoning (since it leads to a wrong result, there are
must be an error!).
The series 1 − 21 + 13 − 14 + 51 + ... = s converges. Its sum can be estimated roughly as 21 < s < 1
(using the first two terms). Now, 2s = 21 − 11 + 32 − 12 + 25 − 13 + 27 − 14 + .... Combining the
terms with odd denominators and leaving those with the even ones, we get
( 21 − 11 ) − 21 + ( 23 − 13 ) − 41 + ( 25 − 15 ) − ... = 1 − 21 + 13 − 14 + 15 − ... = s, so 2s = s but this is
impossible since s 6= 0, s 6= 2. What is the mistake?