So sánh 1/2 ! + 2/ 3! + 3/ 4!

\(\frac{1}{{2!}} + \frac{2}{{3!}} + \frac{3}{{4!}} + \ldots + \frac{{2023}}{{2024!}} = \frac{{2 - 1}}{{2!}} + \frac{{3 - 1}}{{3!}} + \frac{{4 - 1}}{{4!}} + \ldots + \frac{{2024 - 1}}{{2024!}}\)

\( = \frac{2}{{2!}} - \frac{1}{{2!}} + \frac{3}{{3!}} - \frac{1}{{3!}} + \frac{4}{{4!}} - \frac{1}{{4!}} + \ldots + \frac{{2024}}{{2024!}} - \frac{1}{{2024!}}\)

\( = \frac{1}{{1!}} - \frac{1}{{2!}} + \frac{1}{{2!}} - \frac{1}{{3!}} + \frac{1}{{3!}} - \frac{1}{{4!}} + \ldots + \frac{1}{{2023!}} - \frac{1}{{2024!}}\)

\( = 1 - \frac{1}{{2024!}} < 1.\) Vậy \(\frac{1}{{2!}} + \frac{2}{{3!}} + \frac{3}{{4!}} + \ldots + \frac{{2023}}{{2024!}} < 1.\)