Faculty of Engineering and Applied Science

ENGR 1405 Engineering Mathematics 1
2000 Fall

[Note:   some browsers do not display symbols correctly.   Read


“®” as “®” (goes to),	“¥” as “¥” (infinity),
“Þ” as “Þ” (implies),	“ú” as “ú” (Real),
“Ö” as “Ö” (square root),	“-” as “-” (minus sign).]

Some shortcuts for testing series convergence

The full set of tests for series convergence can be found in Dr. Niefer’s notes.
In brief, they are:

• divergence test [cannot prove convergence]
• alternating series test [cannot distinguish between absolute and conditional convergence]
• limit comparison test [best for   a_n = algebraic functions]
• comparison test
• ratio test [first choice for power series; for   u_n containing exponential or factorial factors]
• root test [for   u_n = [f (n)]ⁿ]
• integral test [when f (n) is positive, continuous and decreasing]

Comparison tests usually employ p series or geometric series as the reference series b_n.

A shortcut for determining the limit of f (n) as n ® ¥:

The following types of function form a hierarchy for speed of divergence to infinity.   From greatest dominance to least, they are

nⁿ
n !
cⁿ   (c > 1)
n^d   (d > 0)
log _k n   (k > 1)

Examples:

• n³ / (1.01)ⁿ ® 0   as n ® ¥  because the exponential term is dominant.
• xⁿ / n ! ® 0   as n ® ¥  because the factorial term is dominant.

A shortcut for algebraic functions

a_n is an algebraic function of n if
a_n = (some root of some polynomial) / (a root of some other polynomial)

If   a_n = an algebraic function of n, then the [limit] comparison test leads to a quick test for convergence:
Pick off the highest power of n in the numerator and in the denominator and take the appropriate roots.
The series will converge absolutely if and only if

[overall order in the numerator] < [overall order in the denominator] - 1

Equivalently, compare the overall order 1 / n ^p to that of a p-series.

Examples:

• a_n = (1 + Ön) / (1 + n²):
  Overall order in the numerator = 1/2
  Overall order in the denominator = 2
  Difference = 1/2 - 2 < - 1
  or
  Overall order = 1 / n^3/2 and 3/2 > 1
  Therefore the series converges absolutely.

• a_n = (1 + Ön) / ³Ö(1 + n²):
  Overall order in the numerator = 1/2
  Overall order in the denominator = 2/3
  Difference = 1/2 - 2/3 > - 1
  or
  Overall order = 1 / n^1/6 and 1/6 < 1
  Therefore the series diverges.

  [However, the alternating series
  u_n = (-1)ⁿ (1 + Ön) / ³Ö(1 + n²)
  converges conditionally by “comparison” with the alternating p-series (-1)ⁿ / n^1/6,   (0 < 1/6 <= 1).]