Analysis - Integrability of a Function and Riemann Integral

Описание к видео Analysis - Integrability of a Function and Riemann Integral

Q1: Let f be an integrable function on [a, b] and g a function obtained by changing the values of f for a finite number of points. Show that the function g is integrable on [a, b] and \int_a^b f(x) dx = \int_a^b g(x) dx.

Q2: Let f be an integrable function on [a, b]. Show that
lim_{n to infinity} ((b-a)/n) sum_{i=1}^n f(a + (i-alpha) (b-a)/n) = \int_a^b f(x) dx 0 \le alpha \le 1.

Q3: Show that
lim_{n to infinity} \sum_{k=n}^{2n} 1/k = log 2.

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