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The value of $\int_{0}^{1} lim_{n \to \infty} \sum_{k = 0}^{n} \frac{x^{k + 2} 2^{k}}{k!} d x$ is

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Correct option is D

As limn→∞ ∑k=0n xk+22kk!=x2∑k=0∞ (2x)kk!=x2e2xThus, ∫01 limn→∞ ∑k=0n xk+22kk!dx=∫01 x2e2xdx=142x2−2x+1e2x=14e2−1

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The value of ∫01 limn→∞ ∑k=0n xk+22kk!dx is

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The value of $\int_{0}^{1} lim_{n \to \infty} \sum_{k = 0}^{n} \frac{x^{k + 2} 2^{k}}{k!} d x$ is