Question-197660

Tinku Tara September 25, 2023 Limits 0 Comments

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Question Number 197660 by pticantor last updated on 25/Sep/23

Answered by witcher3 last updated on 25/Sep/23

U_n =∫_0 ^1 (dt/(1+t+...t^n ))≤∫_0 ^1 dt=1 lim_(n→∞) U_n =∫_0 ^1 lim_(n→∞) (dt/(1+...+t^n ))=∫_0 ^1 lim_(n→∞) ((1−t)/(1−t^(n+1) ))dt =∫_0 ^1 1−tdt=(1/2)

U_{n} = \int_{0}^{1} \frac{dt}{1 + t + \dots t^{n}} ⩽ \int_{0}^{1} dt = 1

Double subscripts: use braces to clarify

= \int_{0}^{1} 1 - tdt = \frac{1}{2}

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