Sample Midterm Problems Math 1220
Fall 2010
Instructor: Rémi Lodh
November 4, 2010
1. Find
R0
−∞
x
dx.
(x2 + 1)3/2
R ∞ dx
converges or diverges (hint:
1 xp
consider the cases p < 1, p = 1, and p > 1 separately).
3
cos (n)
.
3. Use the squeeze theorem to find the limit of the sequence
n2
4. Show that if a sequence {an } satisfies limn→∞ |an | = 0, then limn→∞ an = 0.
5. Use the monotonic sequence theorem to show that the sequence 1 − 21 , (1 −
1
1 1
1 1
1
1 1
2 )( 2 − 3 ), (1 − 2 )( 2 − 3 )( 3 − 4 ), ... converges.
P∞
1
6. Find a formula for the nth partial sum of the series k=1
, then find
k(k + 1)
1
the sum of the series (hint: first find the partial fraction decomposition of
).
k(k
+ 1)
k
k+1
P∞
7. Find the sum of the series k=1 2 31 + 5 12
.
P∞
k
8. Determine whether the series k=1
converges or diverges.
k+1
P∞ 1
9. Let p be any number. Determine whether k=1 p converges or diverges (hint:
k
consider the cases p < 1, p = 1, and p > 1 separately).
10. Give an upper bound for the error made by using the 15th partial sum of the
P∞ 1
series k=1 3 to approximate the sum.
k
P∞ ln(k)
11. Determine whether k=1 2 converges or diverges.
k
P∞ 1 + k 2
12. Determine whether k=1
converges or diverges.
k3
P∞
k
converges or diverges.
13. Determine whether k=1 √
k3 + 1
P∞ 5k
14. Determine whether k=1
converges or diverges.
k!
P∞ (−1)k+1
15. Determine whether k=1
converges or diverges.
ln(k)
P∞ (−1)k
16. How many terms of the series k=1
must one add in order to obtain
k3
1
an approximation to within 125 of the sum of the series?
17. Give an example of a series which converges absolutely, a series which converges
conditionally, and a series which diverges.
2. Let p be any number. Determine whether
1
18. Determine whether
diverges.
P∞ (−1)k+1 3k
converges absolutely, conditionally, or
k=1
k!
P∞ (−1)k+1
19. Determine whether k=1 √
converges absolutely, conditionally, or dik
verges.
P∞ cos(k 3 )
20. Show that k=1
converges.
k3
P∞
21. Find the convergence set of the power series n=0 xn . What function does
this series represent on its convergence set?
P∞ (x − 2)n
22. Find the convergence set of the power series n=0
.
n+1
n
P∞ (x + 2)
23. Find the convergence set of the power series n=0
.
3n
24. Find the Taylor series of f (x) = sin(x) at x = π.
25. Use the binomial series expansion of (1 + x)1/3 up to term x3 to approximate
1
√
.
3
3
26. Find the Taylor series expansion of f (x) = sin−1 (x) at x = 0.
2