Fermi
·
Concept Library
·
Courses
·
Books
·
Problems
·
Simulations
·
Math Wiki
·
High School
Fermi
›
Problems
Mathematics
Mathematics
Problem
The geometric series
∑
n
=
1
∞
1
3
n
\sum_{n=1}^{\infty}\dfrac{1}{3^{n}}
∑
n
=
1
∞
3
n
1
is known to converge to
1
2
\tfrac12
2
1
. Consider the regrouped series
(
1
3
+
1
9
)
+
(
1
27
+
1
81
)
+
⋯
(\tfrac13+\tfrac19)+(\tfrac1{27}+\tfrac1{81})+\cdots
(
3
1
+
9
1
)
+
(
27
1
+
81
1
)
+
⋯
. Does this regrouped series converge, and if so, to what value? Give a brief reason for your answer.
Browse the problem library