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SEEMOUS
2023 SEEMOUS
P2
Complicated looking limit
Complicated looking limit
Source: 17th SEEMOUS 2023, Problem 2
March 9, 2023
real analysis
limit
Sequences
Seemous
Problem Statement
For the sequence
S
n
=
1
n
2
+
1
2
+
1
n
2
+
2
2
+
⋯
+
1
n
2
+
n
2
,
S_n=\frac{1}{\sqrt{n^2+1^2}}+\frac{1}{\sqrt{n^2+2^2}}+\cdots+\frac{1}{\sqrt{n^2+n^2}},
S
n
=
n
2
+
1
2
1
+
n
2
+
2
2
1
+
⋯
+
n
2
+
n
2
1
,
find the limit
lim
n
→
∞
n
(
n
⋅
(
log
(
1
+
2
)
−
S
n
)
−
1
2
2
(
1
+
2
)
)
.
\lim_{n\to\infty}n\left(n\cdot\left(\log(1+\sqrt{2})-S_n\right)-\frac{1}{2\sqrt{2}(1+\sqrt{2})}\right).
n
→
∞
lim
n
(
n
⋅
(
lo
g
(
1
+
2
)
−
S
n
)
−
2
2
(
1
+
2
)
1
)
.
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