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Brazil Undergrad MO
2022 Brazil Undergrad MO
4
4
Part of
2022 Brazil Undergrad MO
Problems
(1)
Convergence of series Brazilian MO 2022 P4
Source: Brazilian Mathematics Olympiad 2022 #4
11/22/2022
Let
α
,
c
>
0
\alpha, c > 0
α
,
c
>
0
, define
x
1
=
c
x_1 = c
x
1
=
c
and let
x
n
+
1
=
x
n
e
−
x
n
α
x_{n + 1} = x_n e^{-x_n^\alpha}
x
n
+
1
=
x
n
e
−
x
n
α
for
n
≥
1
n \geq 1
n
≥
1
. For which values of
β
\beta
β
does
∑
i
=
1
∞
x
n
β
\sum_{i = 1}^{\infty} x_n^\beta
∑
i
=
1
∞
x
n
β
converge?
series
real analysis