Use series to approximate numerical value

A question is this type if and only if it asks to substitute a specific value into a series to approximate a number (not an integral).

3 questions

OCR Further Pure Core 2 2021 November Q10
10 In this question you must show detailed reasoning.
  1. By using an appropriate Maclaurin series prove that if \(x > 0\) then \(\mathrm { e } ^ { x } > 1 + x\).
  2. Hence, by using a suitable substitution, deduce that \(\mathrm { e } ^ { t } > \mathrm { e } t\) for \(t > 1\).
  3. Using the inequality in part (b), and by making a suitable choice for \(t\), determine which is greater, \(\mathrm { e } ^ { \pi }\) or \(\pi ^ { \mathrm { e } }\).
SPS SPS FM Pure 2022 February Q12
12. In this question you must show detailed reasoning.
  1. By using an appropriate Maclaurin series prove that if \(x > 0\) then \(\mathrm { e } ^ { x } > 1 + x\).
  2. Hence, by using a suitable substitution, deduce that \(\mathrm { e } ^ { t } > \mathrm { e } t\) for \(t > 1\).
  3. Using the inequality in part (b), and by making a suitable choice for \(t\), determine which is greater, \(\mathrm { e } ^ { \pi }\) or \(\pi ^ { \mathrm { e } }\).
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
    [0pt] [BLANK PAGE]
SPS SPS FM 2024 November Q4
4. (a) (i) Given that \(f ( x ) = \sqrt { 1 + 2 x }\), find \(f ^ { \prime } ( x )\) and \(f ^ { \prime \prime } ( x )\).
(ii) Hence, find the first three terms of the Maclaurin series for \(\sqrt { 1 + 2 x }\).
(b) Hence, using a suitable value for \(x\), show that \(\sqrt { 5 } \approx \frac { 143 } { 64 }\).
[0pt] [BLANK PAGE]