Download 1. Convergence

Series
1. Convergence
Problem 1. Let ak be a sequence of positive numbers. Define sn =
(i)
n
X
ak . Prove that
k=1
∞
X
an
converges.
s2
n=1 n
(ii) If (sn ) is divergent then
∞
X
an
is divergent.
s
n=1 n
Problem 2. Prove that if the series
∞
X
|xn − xn−1 | is convergent then the sequence (xn )n is convergent. Is
n=2
the reciprocal satisfied?
Problem 3. Consider the sequences (an )n , (bn )n , (cn )n , such that an ≤ bn ≤ cn , for all n. If the series
and
∞
X
cn are convergent then the series
n=1
∞
X
∞
X
an
n=1
bn is also convergent.
n=1
Problem 4. Let (pn ) be the increasing sequence of prime positive integers. Prove that the series
∞
X
1
is
p
n=1 n
divergent.
Problem 5. Let (an ) be a decreasing sequence of positive numbers such that the series
∞
X
an is convergent.
n=1
Then lim nan = 0.
n→∞
Solution : Let sn =
n
X
ak . Since (sn )n is convergent, (s[n/2] )n is also convergent with the same limit. Then
k=1
0 ≤ nan ≤ (n − [n/2])an ≤
n
X
n→∞
ak = sn − s[n/2] −→ 0 finishes the proof.
k=[n/2]+1
Problem 6. Let (an ) be a decreasing sequence of positive terms such that lim
n→∞
∞
X
2n a2n
< 1. Then the series
an
an is convergent.
n=1
Problem 7. Let
∞
X
xn be a convergent series with positive terms. Prove that
n=1
x1 + 2x2 + ... + nxn
= 0.
n
(ii) lim nxn = 0
∞
∞
X
x1 + 2x2 + ... + nxn X
=
xn
(iii)
n(n + 1)
n=1
n=1
(i) lim
an + |an |
an − |an |
Problem 8. Given a sequence (an ) we define a+
and a−
.
n = max(an , 0) =
n = min(an , 0) =
2
2
∞
∞
∞
X
X
X
(a) If
an is absolutely convergent, show that both of the series
a+
a−
n and
n are convergent.
n=1
(b) If
∞
X
n=1
an is conditionally convergent, show that both of the series
n=1
n=1
Problem 9. Prove that if
rearrangement of
∞
X
∞
X
n=1
∞
X
n=1
a+
n and
∞
X
a−
n are divergent.
n=1
an is a conditionally convergent series and r is a real number, than ther is a
n=1
an whose sum is r.
2
Problem 10. Consider the series S whose terms are the reciprocals of the positive integers that can be written
in base 10 notation without using the digit 0. Show that this series is convergent and the sum is less than 90.
X
an+1
Problem 11. Let
an be a series with positive terms. Suppose that the sequence (rn ) defined by rn =
an
∞
X
X
is convergent and l = lim rn < 1, so
an converges. Let Rn =
ak .
n→∞
k=n+1
an+1
(a) If (rn ) is a decresing sequence and rn+1 < 1, show that Rn ≤
1 − rn+1
an+1
(b) If (rn ) is an increasing sequence, show that Rn ≤
.
1−l
X
Problem 12. If
an is a convergent series with positive terms, is it true that
X
(a)
ln(1 + an ) is convergent?
X
(b)
sin(an ) is convergent?
X
(c)
a2n is also convergent?
X
X
(d)
an bn is also convergent, where
bn is a convergent series with positive terms?
X
Problem 13. Show that if an > 0 and lim nan 6= 0, then
an is divergent.
x
P∞
Problem 14. Determine, with proof, the set of real numbers x for which n=1 n1 csc n1 − 1 converges.
P∞
P∞
n
Problem 15. Prove that if n=1 an is a convergent series of positive real numbers, then so is n=1 (an ) n+1 .
P∞Problem 16. Let (an ) be a sequence of positive reals such that, for all n, an ≤ a2n + a2n+1 . Prove that
n=1 an diverges. [P1994]
X
Problem 17. Let (an ) be a sequence of positive numbers such that
an converges. Find a necessary and
X an
X
sufficient condition for the existence of a sequence of positive numbers (bn ) such that
and
bn both
bn
converge.
Problem 18. Let a1 , a2 , ..., an be positive numbers such that
numbers c1 , c2 , ... such that lim cn = ∞ and
n→∞
∞
X
∞
X
an is convergent. Prove that there are positive
n=1
cn an is convergent.
n=1
2. Sums
∞
X
Problem 19. Evaluate S =
p=2
Problem 20. Evaluate S =
∞
X
k=1
!
∞
1X 1
.
p q=2 q p
1
.
2k 2 + 3k
Problem 21. A sequence (an ) is defined recursively by the equations a0 = a1 = 1 and n(n − 1)an = (n −
∞
X
1)(n − 2)an−1 − (n − 3)an−2 . Find the sum of the series
an .
n=0
Problem 22. If p > 1, evaluate the expression
1
1
1
1 + p + p + + p ...
2
3
4
1
1
1
1 − p + p − p ...
2
3
4
Problem 23. Find the sum of the series
1 1 1 1 1 1
1
1+ + + + + + +
...
2 3 4 6 8 9 12
where the terms are the reciprocals of the positive integers whose only prime factors are 2s and 3s.
Problem 24. Find the sum of the series
∞
X
1
x
tan n .
n
2
2
n=1
2. SUMS
3
Problem 25. Let B(n) be the number of ones that appear in the base two expression for the positive integer
∞
X
B(n)
n. Evaluate
.
n(n + 1)
n=1
Problem 26. For x > 0 evaluate
Problem 27. Evaluate S =
n
∞
X
(−1)[2 x]
.
2n
n=1
∞ X
∞
X
n=1m=1
m2 n
1
+ mn2 + 2mn
Problem 28. For x > 1 determine the sum of the series
n
∞
X
x2
S(x) =
(x + 1)(x2 + 1) . . . (x2n + 1)
n=0
Problem 29. Let x > 1. Find the sum of the series S =
∞
X
n=0
Problem 30. Evaluate
∞
X
n=0
Problem 31. Evaluate
2n
.
+1
x2n
cos n
.
2n
∞
X
n2 + 3n + 1
.
(n2 + n)2
n=1
Problem 32. Evaluate the series
∞
X
n=0
n
.
(n + 1)!
Problem 33. If (fn ) is the Fibonacci sequence defined by f0 = 1, f1 = 1 and fn+1 = fn + fn−1 , evaluate the
∞
∞
X
X
1
fn
series
and
.
f
f
f
f
n=1 n−1 n+1
n=1 n−1 n+1
Problem 34. Let k ≥ 2 be a fixed integer. For n ≥ 1 define an = 1, if n is not a multiple of k, and an = 1 − k,
∞
X
an
.
if n is a multiple of k. Evaluate
n
n=1
Problem 35. Evaluate
Problem 36. Evaluate
∞
X
n2
.
n!
n=0
∞
X
n=1
Problem 37. Evaluate
∞
X
n=0
Problem 38. Let an =
6n
(3n+1 − 2n+1 )(3n − 2n )
.
n
n4 + n2 + 1
∞
X
1
1
1
+
−
, n = 1, 2, . . .. Does the series
an converge, and if so,
4n + 1
4n + 3
2n + 2
n=0
what is its sum?
Problem 39. Evaluate
∞
X
(−1)n−1
.
n
n=1
Problem 40. Evaluate the infinite series S =
∞
X
(−1)n
n=0
Problem 41. Evaluate the infinite series
∞
X
n=1
arctan
(n + 1)3
n!
2
.
n2
P∞
Problem 42. Let A be a positive real number. What are the possible values of j=0 x2j , given that x0 , x1 , . . .
P∞
are positive numbers for which j=0 xj = A? [P2000]
P∞
Problem 43. Evaluate n=0 Arccot(n2 + n + 1), where Arccot t for t ≥ 0 denotes the number θ in the interval
0 < θ ≤ π/2 with cot θ = t. [P1986]
4
Problem 44. Evaluate
∞ X
∞
X
m=1n=1
m2 n
.
3m (n3m + m3n )
Problem 45. For any positive integer n, let hni denote the closest integer to
√
n. Evaluate
∞
X
2hni + 2−hni
.
2n
n=1