ENGR 1405 Engineering Mathematics 1

Faculty of Engineering and Applied Science
2000 Fall

Problem Set 7
Questions

  1. Determine whether the series Sum(n=0 to oo) { sin n / exp n } is absolutely convergent, conditionally convergent or divergent.   State which test you are using.

  1. Determine whether the series Sum(n=1 to oo) { ((sin n)/n)^n } is absolutely convergent, conditionally convergent or divergent.   State which test you are using.

  1. Find the range(s) of values of   z   for which the series Sum(n=1 to oo) { (1/n)*((z-1)/(z+1))^n } is
    (i)absolutely convergent,
    (ii)conditionally convergent,
    (iii)divergent.
    State which test you are using.

  1. For the function   f (x) = cos x:

    1. Find the Taylor series for   f (x)   about the centre   c = p/3 .  
    2. Express this Taylor series as a linear combination of Maclaurin series expansions of   cos (x - p/3)   and   sin (x - p/3).
    3. Find the interval of convergence of this Taylor series.
    4. Hence verify that the trigonometric identity
              cos (A + B) = cos A cos B - sin A sin B
      is valid when   A = p/3   and   B = x - p/3.
  1.   f (x) = cos (sqrt(x)) ;   in powers of "x".
  1.   f (x) = cosh (2x) = ½(e2x + e-2x) ;   in powers of "x".
  1.   f (x) = cube root of {2 - x} about the centre   c = 1.
  1.   f (x) = 1 / (x-4)^2;   in powers of "x - 2".

  1. Determine the first five (5) non-zero terms of the Maclaurin series expansion for

    1. f (x) = tan (x)
    2. f (x) = ln [cos (x)]

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