Guide to Infinite Series Tests

Table of Tests to use on Infinite Series: (see notes below)

 Test Series Convergence or Divergence Comments nth Term Diverges if Inconclusive if lim = 0 GeometricSeries (i) Converges if | r | < 1(ii) Diverges if | r | 1 If convergent, the sum is Useful for comparison tests p-series (i) Converges if p > 1(ii) Diverges if p 1 Useful for comparison tests Integral an = f(x) (i) Converges if converges(ii) Diverges if diverges f (x) must be continuous, positive, decreasing. Comparison TestLimit Comparison Test test , select both positive (i) converges if converges and for all values of n.(ii) diverges if diverges and for all values of n.(iii) If , then both series converge or both series diverge. The comparison series is often a geometric or a p-series. Ratio Test If ),:(i) converges if L < 1(ii) diverges if L > 1 Inconclusive if L = 1.Useful if an involves factorials or nth powersIf all a are > 0, ignore the absolute value signs. Root Test If ,the series:(i) converges if L < 1(ii) diverges if L > 1 Inconclusive if L = 1.Useful if an includes only nth powersIf all a's are > 0, ignore the absolute value sign. Alternating Series Test an > 0 Converges if ak a k+1 for every kand Applies only to an alternating series. Absolute Convergence Test If is convergent then is convergent Useful for series that contain both positive & negative terms.

____________________________________________

Notes:

Because there are so many different tests, this topic can be a major brain-strain unless we develop some strategies to approach the questions! Here then is a list of hints that might make things a little easier.

1) if the limit at inifinty is easy to find, run the nth term test for divergence.
If the limit isn't = 0, the series Diverges. Ex:
after we divide all terms in the fraction by n², the limit is the ratio of the coefficients.
2) remember the binary relation (< , = , or > ) between 2 fractions:
because n² + 3 is greater than n², so the quotient is smaller.
use this approach for basic comparison test.
3) Learn to recognize the formats for certain families of series;
Geometric series: with constant base and n as exponent,
p-series: with n as the base and p a constant exponent,
telescoping series: converges to 1
harmonic series to use with the comparison tests.
is a mutation of the harmonic series , and so diverges.
4) Recognize an Alternating Series even if we don't see ( – 1) n in the term;
once we factor it.

(all content © MathRoom Learning Service; 2004 - ).