MathDB
Problems
Contests
Undergraduate contests
IMC
1997 IMC
2
sum convergence
sum convergence
Source: IMC 1997 day 1 problem 2
October 1, 2005
real analysis
real analysis unsolved
Problem Statement
Let
a
n
a_n
a
n
be a sequence of reals. Suppose
∑
a
n
\sum a_n
∑
a
n
converges. Do these sums converge aswell? (a)
a
1
+
a
2
+
(
a
4
+
a
3
)
+
(
a
8
+
.
.
.
+
a
5
)
+
(
a
16
+
.
.
.
+
a
9
)
+
.
.
.
a_1+a_2+(a_4+a_3)+(a_8+...+a_5)+(a_{16}+...+a_9)+...
a
1
+
a
2
+
(
a
4
+
a
3
)
+
(
a
8
+
...
+
a
5
)
+
(
a
16
+
...
+
a
9
)
+
...
(b)
a
1
+
a
2
+
(
a
3
)
+
(
a
4
)
+
(
a
5
+
a
7
)
+
(
a
6
+
a
8
)
+
(
a
9
+
a
11
+
a
13
+
a
15
)
+
(
a
10
+
a
12
+
a
14
+
a
16
)
+
(
a
17
+
a
19
+
.
.
.
{a_1+a_2+(a_3)+(a_4)+(a_5+a_7)+(a_6+a_8)+(a_9+a_{11}+a_{13}+a_{15})+(a_{10}+a_{12}+a_{14}+a_{16})+(a_{17}+a_{19}+...}
a
1
+
a
2
+
(
a
3
)
+
(
a
4
)
+
(
a
5
+
a
7
)
+
(
a
6
+
a
8
)
+
(
a
9
+
a
11
+
a
13
+
a
15
)
+
(
a
10
+
a
12
+
a
14
+
a
16
)
+
(
a
17
+
a
19
+
...
Back to Problems
View on AoPS