Crazy Limits! 1

Calculus Level 2

lim ⁡ n → ∞ 1 2 + 2 2 + 3 2 + ⋯ + n 2 n 2 ( n 2 + 1 ) = ? \large \lim_{n \to \infty} \frac{1^2+2^2+3^2+\cdots+n^2}{n^2(n^2+1)} = \, ?


The answer is 0.

Shubhang Mundra by Shubhang Mundra , 20, India

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2 solutions

First Last
Jan 11, 2016

The Numerator is clearly ∑ i = 1 n ( i 2 ) \sum_{i=1}^n (i^2) which is equal to n ( n + 1 ) ( 2 n + 1 ) 6 \frac{n(n+1)(2n+1)}{6}

Rewrite this as n × n × ( 1 + 1 n ) ( 2 + 1 n ) n \times n \times (1+\frac{1}{n})(2+\frac{1}{n})

This n^2 cancels out with the denominator, and as n --> ∞ the Numerator becomes a constant and the Denominator becomes infinite so the limit equals 0 \boxed{0}

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Righved K
Jan 7, 2016

Did this problem by sandwich theorum:D

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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