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
Geometric Series		(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 Test Limit 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 powers If 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 powers If all a's are > 0, ignore the absolute value sign.
Alternating Series Test	an > 0	Converges if ak a k+1 for every k and	Applies only to an alternating series.
Absolute Convergence Test		If is convergent then is convergent	Useful for series that contain both positive & negative terms.

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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.

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