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Problems
Let
Y
n
(
x
)
=
x
2
+
x
2
1
+
x
2
+
x
2
(
1
+
x
2
)
2
+
⋯
+
x
2
(
1
+
x
2
)
n
−
1
Y_n(x) = x^2 + \frac{x^2}{1+x^2} + \frac{x^2}{(1+x^2)^2} + \dots + \frac{x^2}{(1+x^2)^{n-1}}
Y
n
(
x
)
=
x
2
+
1
+
x
2
x
2
+
(
1
+
x
2
)
2
x
2
+
⋯
+
(
1
+
x
2
)
n
−
1
x
2
…
Mathematics
Problem
Let
Y
n
(
x
)
=
x
2
+
x
2
1
+
x
2
+
x
2
(
1
+
x
2
)
2
+
⋯
+
x
2
(
1
+
x
2
)
n
−
1
Y_n(x) = x^2 + \frac{x^2}{1+x^2} + \frac{x^2}{(1+x^2)^2} + \dots + \frac{x^2}{(1+x^2)^{n-1}}
Y
n
(
x
)
=
x
2
+
1
+
x
2
x
2
+
(
1
+
x
2
)
2
x
2
+
⋯
+
(
1
+
x
2
)
n
−
1
x
2
and
Y
(
x
)
=
lim
n
→
∞
Y
n
(
x
)
Y(x) = \lim_{n \to \infty} Y_n(x)
Y
(
x
)
=
lim
n
→
∞
Y
n
(
x
)
. Discuss the continuity of
Y
n
(
x
)
Y_n(x)
Y
n
(
x
)
(
n
=
1
,
2
,
3
,
…
n = 1, 2, 3, \dots
n
=
1
,
2
,
3
,
…
) and
Y
(
x
)
Y(x)
Y
(
x
)
at
x
=
0
x = 0
x
=
0
.
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