The Convergence Theorem for Power Series

There are three possibilities for

cn ( x  a ) n with respect to
n 0
convergence:
1. There is a positive number R such that the series diverges
x  a  R but converges for x  a  R . The series
may or may not converge at either of the endpoints x  a  R
and x  a  R.
for
The Convergence Theorem for Power Series
(Continued).
2. The series converges for every x ( R  ).
3. The series converges at
elsewhere
( R  0).
xa
and diverges
The nth-Term Test for Divergence

a
n
n 1
diverges if lim an fails to
n 
exist or is different from zero.
The Direct Comparison Test
Let
a
(a)
a
n
n
series
be a series with no negative terms.
converges if there is a convergent
c
n
with
an  cn
for all
n  N,
for some integer N.
(b)
a
n
diverges if there is a divergent series
of nonnegative terms with
an  d n
n  N , for some integer N.
for all
d
n
Absolute Convergence
If the series  an of absolute value converges, then
a
n
converges absolutely.
Absolute Convergence Implies Convergence
If
a
n
converges, then  an converges.
The Ratio Test
Let
a
n
be a series with all positive terms,
and with
an 1
lim
 L.
n  a
n
Then,
(a) The series converges if L < 1,
(b) The series diverges if L > 1,
(c) The test is inconclusive if L = 1.
The Integral Test
Let
an 
be a sequence of positive terms.
an  f (n) , where f is a continuous,
positive, decreasing function of x for all x  N

Suppose that
(N a positive integer). Then the series

a
n N
n
and
the integral
 f ( x) dx either both converge or both
diverge.
N
The p-Series Test

1.
 (1/ n
P
2.
 (1/ n
P
) diverges if p < 1.
P
) diverges if p = 1. (Harmonic Series)
) converges if p > 1.
n 1

n 1

3.
 (1/ n
n 1
The Limit Comparison Test (LCT)
Suppose that
an  0
and
bn  0 for all n  N
(N a
positive integer).
an
 c, 0  c  , then
1. If lim
n b
n
a
and
b
an
 0 and  bn converges, then
2. If lim
n b
n
a
n
n
both
converge or both diverge.
an
  and
3. If lim
n b
n
b
n
diverges, then
n
a
n
converges.
diverges.
The Alternating Series Test (Leibniz’s Theorem)
The series

 (1)
n 1
n 1
un  u1  u2  u3  u4  ...
converges if all three of the following conditions are satisfied:
1. each
un
is positive.
un  un1 for all n  N, for some integer N.
3. lim un  0.
2.
n
The Alternating Series Estimation Theorem

n 1
(

1)
un satisfies the
If the alternating series 
n 1
conditions of Leibniz’s Theorem, then the truncation error
for the nth partial sum is less than
sign as the first unused term.
un1 and has the same
nth-root Test