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Undergraduate contests
IMC
1996 IMC
11
11
Part of
1996 IMC
Problems
(1)
IMC 1996 Problem 11
Source: IMC 1996
3/4/2021
i) Prove that
lim
x
→
∞
∑
n
=
1
∞
n
x
(
n
2
+
x
)
2
=
1
2
\lim_{x\to \infty}\,\sum_{n=1}^{\infty} \frac{nx}{(n^{2}+x)^{2}}=\frac{1}{2}
x
→
∞
lim
n
=
1
∑
∞
(
n
2
+
x
)
2
n
x
=
2
1
. ii) Prove that there is a positive constant
c
c
c
such that for every
x
∈
[
1
,
∞
)
x\in [1,\infty)
x
∈
[
1
,
∞
)
we have
∣
∑
n
=
1
∞
n
x
(
n
2
+
x
)
2
−
1
2
∣
≤
c
x
\left|\sum_{n=1}^{\infty} \frac{nx}{(n^{2}+x)^{2}}-\frac{1}{2} \right| \leq \frac{c}{x}
n
=
1
∑
∞
(
n
2
+
x
)
2
n
x
−
2
1
≤
x
c
limit
real analysis