

Definition of Convergence and Divergence in Series
The n^{th} partial sum of the series
a_{n} is given by S_{n} = a_{1} + a_{2}
+ a_{3} + ... + a_{n}. If the sequence of these partial
sums {S_{n}} converges to L, then the sum of the series converges
to L. If {S_{n}} diverges, then the sum of the series diverges.
Operations on Convergent Series
If a_{n}
= A, and b_{n}
= B, then the following also converge as indicated:
ca_{n} = cA
(a_{n}
+ b_{n}) = A + B
(a_{n}
 b_{n}) = A  B
Alphabetical Listing of Convergence Tests
Absolute Convergence
If the series
a_{n} converges, then the series
a_{n} also converges.
Alternating Series Test
If for all n, a_{n} is positive, nonincreasing (i.e.
0 < a_{n+1} <= a_{n}), and approaching zero, then
the alternating series
(1)^{n}
a_{n} and
(1)^{n1} a_{n}
both converge.
If the alternating series converges, then the remainder R_{N}
= S  S_{N} (where S is the exact sum of the infinite series and
S_{N} is the sum of the first N terms of the series) is bounded
by R_{N} <= a_{N+1}
Deleting the first N Terms
If N is a positive integer, then the series
both converge or both diverge.
Direct Comparison Test
If 0 <= a_{n} <= b_{n} for all n greater
than some positive integer N, then the following rules apply:
If b_{n}
converges, then
a_{n} converges.
If a_{n}
diverges, then
b_{n} diverges.
Geometric Series Convergence
The geometric series is given by
a r^{n} = a + a r + a r^{2} + a r^{3} + ...
If r < 1 then the following geometric series converges to a / (1
 r).
If r >= 1 then the above geometric series diverges.
Integral Test
If for all n >= 1, f(n) = a_{n}, and f is positive,
continuous, and decreasing then
either both converge or both diverge.
If the above series converges, then the remainder R_{N} = S 
S_{N} (where S is the exact sum of the infinite series and S_{N}
is the sum of the first N terms of the series) is bounded by 0< = R_{N}
<= (N..)
f(x) dx.
Limit Comparison Test
If lim (n>) (a_{n}
/ b_{n}) = L,
where a_{n}, b_{n} > 0 and L is finite and positive,
then the series
a_{n} and
b_{n} either both converge or both diverge.
n^{th}Term Test for Divergence
If the sequence {a_{n}} does not converge to zero, then the series
a_{n} diverges.
pSeries Convergence
The pseries is given by
1/n^{p}
= 1/1^{p} + 1/2^{p} + 1/3^{p} + ...
where p > 0 by definition.
If p > 1, then the series converges.
If 0 < p <= 1 then the series diverges.
Ratio Test
If for all n, n
0, then the following rules apply:
Let L = lim (n  > )
 a_{n+1} / a_{n} .
If L < 1, then the series
a_{n} converges.
If L > 1, then the series
a_{n} diverges.
If L = 1, then the test in inconclusive.
Root Test
Let L = lim (n  > )
 a_{n} ^{1/n}.
If L < 1, then the series
a_{n} converges.
If L > 1, then the series
a_{n} diverges.
If L = 1, then the test in inconclusive.
Taylor Series Convergence
If f has derivatives of all orders in an interval I centered
at c, then the Taylor series converges as indicated:
(1/n!)
f^{(n)}(c) (x  c)^{n} = f(x)
if and only if lim (n>)
RN_{} = 0 for all x in I.
The remainder R_{N} = S  S_{N} of the Taylor series (where
S is the exact sum of the infinite series and S_{N} is the sum
of the first N terms of the series) is equal to (1/(n+1)!) f^{(n+1)}(z)
(x  c)^{n+1}, where z is some constant between x and c.


