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Limit of a sequence (TIFR 2013 problem 32)

Question:

True/False?

\( \lim_{n\to \infty } (n+1)^{1/3} -n^{1/3} = \infty \)

Hint:

Simplify the given expression.

Discussion:

We feel that \( (n+1)^{1/3} \) goes to infinity at the same speed as \( n^{1/3} \). So in fact, the above limit should be zero.

We make this little bit more rigorous.

\(   (n+1)^{1/3} -n^{1/3} = \frac{n+1-n}{(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3} } \)

\(  =\frac{1}{(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3} } \to 0\) as \(n\to \infty \).

 

September 10, 2017

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