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ISI Entrance Examination
2024 ISI Entrance UGB
P1
P1
Part of
2024 ISI Entrance UGB
Problems
(1)
ISI 2024 P1
Source:
5/12/2024
Find, with proof, all possible values of
t
t
t
such that
lim
n
→
∞
(
1
+
2
1
/
3
+
3
1
/
3
+
⋯
+
n
1
/
3
n
t
)
=
c
\lim_{n \to \infty} \left( \frac{1 + 2^{1/3} + 3^{1/3} + \dots + n^{1/3}}{n^t} \right ) = c
n
→
∞
lim
(
n
t
1
+
2
1/3
+
3
1/3
+
⋯
+
n
1/3
)
=
c
for some real
c
>
0
c>0
c
>
0
. Also find the corresponding values of
c
c
c
.
calculus
limit
limit as integration of sum
integration
ISI 2024