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$E v a l u a t e lim_{n \to \infty} \frac{(1^{2} + 2^{2} + 3^{3} + \dots + n^{2}) (1^{3} + 2^{3} + 3^{3} + \dots + n^{3})}{1^{6} + 2^{6} + 3^{6} + \dots + n^{6}}$

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Correct option is C

The given limit islimn→∞ ∑r=1n r2×∑r=1n r3∑r=1n r6=limn→∞ n2∑r=1n rn2×n3∑r=1n rn3n6∑r=1n rn6 =limn→∞ 1n∑r=1n rn2×1n∑r=1n rn31n∑r=1n rn6 =∫01 x2dx∫01 x3dx∫01 x6dx=x3301x4401x7701=13×1417=712

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Evaluate limn→∞ 12+22+33+⋯+n213+23+33+⋯+n316+26+36+⋯+n6

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$E v a l u a t e lim_{n \to \infty} \frac{(1^{2} + 2^{2} + 3^{3} + \dots + n^{2}) (1^{3} + 2^{3} + 3^{3} + \dots + n^{3})}{1^{6} + 2^{6} + 3^{6} + \dots + n^{6}}$