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DPiD Maths Mock Set 1 Paper 2 · Q13

Paper 2Logic & proofCalculusNumber & counting

Let nn be a positive integer and define In=∫01xnex dx\displaystyle I_n=\int_0^1x^ne^x\,dx. Which of the following statements is true?

AI1=1I_1=1
BI1=e−1I_1=e-1
CThe sequence (In)(I_n) is increasing.
DIn→eI_n\to e as n→∞n\to\infty.
Een+1≤In≤1n+1\frac e{n+1}\leq I_n\leq\frac1{n+1}
FI1=eI_1=e
GThe sequence (In)(I_n) does not tend to 00.
HIn≥1I_n\geq1 for all nn.

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