Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 11 / b

Table of contents
- i b
  - Solution i
- ii b
  - Solution ii

i

Solution

Multiply the Maclaurin series for the exponential and the sine, keeping terms through degree three:

e^{x} sin x = (1 + x + \frac{x ^{2}}{2} + \frac{x ^{3}}{6} + O (x^{4})) (x - \frac{x ^{3}}{6} + O (x^{5})) .

Only

x

,

x^{2}

, and

(1/2 - 1/6) x^{3}

contribute at these orders, so

e^{x} sin x = x + x^{2} + \frac{x ^{3}}{3} + O (x^{4}) .

ii

Solution

Put

v = x + x^{2}

in the logarithmic series

ln (1 + v) = v - v^{2} /2 + v^{3} /3 + O (v^{4})

. Since

v^{2} = x^{2} + 2 x^{3} + O (x^{4})

and

v^{3} = x^{3} + O (x^{4})

,

ln (1 + x + x^{2}) = x + x^{2} - \frac{1}{2} (x^{2} + 2 x^{3}) + \frac{1}{3} x^{3} + O (x^{4}) .

Therefore

ln (1 + x + x^{2}) = x + \frac{x ^{2}}{2} - \frac{2 x ^{3}}{3} + O (x^{4}) .

 Ancestors (12)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version