This document summarizes various tests that can be used to determine if an infinite series converges or diverges, including: 1) The divergence test, integral test, p-series test, comparison test, limit comparison test, alternating series test, and ratio test. 2) It also discusses power series, including determining the radius of convergence and using Taylor series approximations with Taylor's inequality to estimate the remainder term. 3) Key concepts are that convergence tests check if partial sums approach a limit, while divergence tests examine the behavior of individual terms, and that power series have a radius of convergence determining the interval on which they converge.

1. MATH 1220 Summary of Convergence Tests for Series
∞
Let ∑a
n=1
n be an infinite series of positive terms.
∞
The series ∑a
n=1
n converges if and only if the sequence of partial sums,
∞
S n = a1 + a 2 + a3 +  a n , converges. NOTE: lim S n = ∑a n
n→∞
n=1
∞
Divergence Test: If n → ∞ an ≠ 0 , the series
lim ∑a n diverges.
n=1
n 1
∞
n lim = lim =1
Example: The series ∑ is divergent since n→ ∞
n2 + 1 n→ ∞
1+ 1
n =1 n +12
n2
This means that the terms of a convergent series must approach zero. That is, if ∑ n
a
converges, then lim an = 0. However, lim a n = 0 does not imply convergence.
n →∞ n →∞
Geometric Series: THIS is our model series A geometric series
a + ar + ar 2 +  + ar n −1 +  converges for − 1 < r < 1 .
an +1 a
Note: r = If the series converges, the sum of the series is .
an 1 −r
n
∞
7  35 7
Example: The series ∑5 8  converges with a = a1 = 8 and r = 8 . The sum of the
n=1  
series is 35.
Integral Test: If f is a continuous, positive, decreasing function on [1, ∞ with
)
∞
f ( n) = an , then the series ∑a
n=1
n converges if and only if the improper integral
∞
∫ f ( x)dx converges.
1
∞
Remainder for Integral Test: If ∑a
n=1
n converges by the Integral Test, then the
∞
remainder after n terms satisfies Rn ≤∫ f ( x)dx
n
∞
1
p-series: The series ∑n
n=1
p is convergent for p > 1 and diverges otherwise.
∞ ∞
1 1
Examples: The series ∑n
n=1
1.001 is convergent but the series ∑n
n=1
is divergent.

2. ∞ ∞
Comparison Test: Suppose ∑a
n=1
n and ∑b
n=1
n are series with positive terms.
∞ ∞
(a) If ∑b
n=1
n is convergent and an ≤ bn for all n, then
n=1
∑a n converges.
∞ ∞
(b) If ∑b
n=1
n is divergent and an ≥ bn for all n, then ∑a
n=1
n diverges.
The Comparison Test requires that you make one of two comparisons:
• Compare an unknown series to a LARGER known convergent series (smaller
than convergent is convergent)
• Compare an unknown series to a SMALLER known divergent series (bigger
than divergent is divergent)
∞ ∞ ∞
3n 3n 1
Examples: ∑n
n =2
2
> ∑ 2 = 3∑ which is a divergent harmonic series. Since the
− 2 n =2 n n =2 n
original series is larger by comparison, it is divergent.
∞ ∞
5n 5n 5 ∞ 1
We have ∑ 2n 3 + n 2 + 1 n=1 2n 2 n=1 n
n =1
< ∑ 3 = ∑ 2 which is a convergent p-series. Since the
original series is smaller by comparison, it is convergent.
∞ ∞
Limit Comparison Test: Suppose ∑a
n=1
n and ∑b
n=1
n are series with positive terms. If
an
lim = c where 0 < c < ∞ , then either both series converge or both series diverge.
n →∞ b
n
(Useful for p-series)
Rule of Thumb: To obtain a series for comparison, omit lower order terms in the
numerator and the denominator and then simplify.
∞ ∞
∞ n 1
Examples: For the series ∑ 2
n
n =1 n + n + 3
, compare to ∑n 2
=∑ 3 which is a
n =1 n =1 n 2
convergent p-series.
πn + n
n
∞ ∞
πn ∞
π 
For the series ∑ n 2 , compare to ∑ n
= ∑  which is a divergent geometric
n =1 3 + n n =1 3 n =1  3 
series.
Alternating Series Test: If the alternating series
∞
∑( −1)
n −1
bn = b1 − b2 + b3 − b4 + b5 − b6 + 
n =1
satisfies (a) bn > bn +1 and (b) n →∞ bn = 0 , then the series converges.
lim
Remainder: Rn = s − sn ≤ bn +1
Absolute convergence simply means that the series converges without alternating (all
signs and terms are positive).
∞
( −1) n
Examples: The series ∑
n =0 n +1
is convergent but not absolutely convergent.
(− ) n
1 ∞
Alternating p-series: The alternating p-series ∑ p converges for p > 0.
n=1 n

3. ∞
( −1) n ∞
( −1) n
Examples: The series ∑
n=1 n
and the Alternating Harmonic series ∑
n=1 n
are
convergent.

4. ∞
an +1 an +1
lim
Ratio Test: (a) If n →∞
an
< 1 then the series ∑a
n=1
n lim
converges; (b) if n →∞
an
>1
the series diverges. Otherwise, you must use a different test for convergence.
This says that if the series eventually behaves like a convergent (divergent) geometric
series, it converges (diverges). If this limit is one, the test is inconclusive and a different
test is required. Specifically, the Ratio Test does not work for p-series.
POWER SERIES
∞
Radius of Convergence: The radius of convergence for a power series ∑c
n =0
n ( x − a) n
cn
is R = n →∞
lim . The center of the series is x = a. The series converges on the open
cn +1
interval ( a − R, a + R ) and may converge at the endpoints. You must test each series
that results at the endpoints of the interval separately for convergence.
∞
( x + 2) n
Example: The series ∑ (n +1)
n =0
2 is convergent on [-3,-1] but the series
∞
(−1) n ( x − 3) n
∑
n =0 5n n +1
is convergent on (-2,8].
Taylor Series: If f has a power series expansion centered at x = a, then the power series
∞
f ( n ) (a)
is given by f ( x) = ∑ ( x − a) n . Use the Ratio Test to determine the interval of
n =0 n!
convergence.
(n + )
Taylor’s Inequality (Remainder): If f
1
( x ) ≤ M for x − ≤d , then
a
M n +1
Rn ( x) ≤ x −a for x −a ≤d . Note that d < R (the radius of convergence)
( n +1)!
( n+ )
1
and think of M as the maximum value of f ( x ) on the interval [x,a] or [a,x].