SOLUTION TO AMM PROBLEM # 11384 Problem# 11384 Proposed

SOLUTION TO AMM PROBLEM # 11384
ÁNGEL PLAZA
Problem# 11384 Proposed by Moubinool Omarjee, Lycée Jean-Lurcat, Paris, France.
Show that
√
∞
X
(−1)⌊ n⌋
(pn
n=1
converges.
Solution. It is enough to apply the Leibnitz’s Theorem for alternating series. Note that
∞
X
(−1)⌊
n=1
pn
√
n⌋
=
∞
X
n
(−1)
n=1
2n + 1
< an =
p(n+1)2 −1
an − an+1 >
X
(n+1)2 −1
k=n2
X
(n+1)2 −1
k=n2
X
1
=
(−1)n an
pk
n=1
∞
1
2n + 1 n→∞
<
−→ 0
pk
pn2
2n + 3
2n + 1
−
>0
p(n+1)2 −1 p(n+1)2
¤
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