MathDB
Problems
Contests
National and Regional Contests
India Contests
India LIMIT
2019 LIMIT
2019 LIMIT Category B
Problem 1
product of sequence and limit
product of sequence and limit
Source: LIMIT 2019 CBS1 P1
April 28, 2021
real analysis
Problem Statement
Let
a
1
=
1
a_1=1
a
1
=
1
and
a
n
=
n
(
a
n
−
1
+
1
)
a_n=n(a_{n-1}+1)
a
n
=
n
(
a
n
−
1
+
1
)
for
n
≥
2
n\ge2
n
≥
2
. Define
p
n
=
∏
i
=
1
n
(
1
+
1
a
i
)
p_n=\prod_{i=1}^n\left(1+\frac1{a_i}\right)
p
n
=
i
=
1
∏
n
(
1
+
a
i
1
)
Then
lim
n
→
∞
p
n
\lim_{n\to\infty}p_n
lim
n
→
∞
p
n
is
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1
+
e
<span class='latex-bold'>(A)</span>~1+e
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
1
+
e
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
e
<span class='latex-bold'>(B)</span>~e
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
e
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
1
<span class='latex-bold'>(C)</span>~1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
∞
<span class='latex-bold'>(D)</span>~\infty
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
∞
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