Math review
O.dubaie
series_convergence.pdf
Infinite Series Convergence Tests
For the Infinite Series
1n
n a , convergence can be evaluated by using the following tests.
Remember, the Geometric and Telescoping Series are the only types of series whose convergence can be calculated exactly.
The other tests serve the purpose of determining whether or not the series converges, not to what finite number it converges.
Test Format Results
Geometric
1
1
n
n ar If r<1, then the series converges to
r
a
1
Telescoping
1 ))((n knhn
a
Expand an by Partial Fraction Decomposition
Canceling may occur, making the convergence obvious by examining n
n s
lim as a
sequence, where sn = a1 + a2 + . . . + an.
p - series
1n p
n
a
If p > 1, the series converges.
If p 1, the series diverges
Integral
1
)( dxxf , where f(n) = an for every n = 1, 2, 3 . . . If the improper integral converges, so does the series.
If the improper integral diverges, so does the series.
Comparison Compare
1n
n a with a series of known convergence,
1n
n b .
If an bn for all n N, and
1n
n b is known to converge, then so does
1n
n a .
If an bn for all n N, and
1n
n b is known to diverge, then so does
1n
n a .
Limit Comparison
Compare
1n
n a with a series of known convergence,
1n
n b ,
by simplifying the ratio
n
n
b
a
If
n
n
n b
a
lim converges to a finite positive number as a sequence, then
1n
n a and
1n
n b
either both converge or both diverge.
Infinite Series Convergence Tests
For the Infinite Series
1n
n a , convergence can be evaluated by using the following tests.
Remember, the Geometric and Telescoping Series are the only types of series whose convergence can be calculated exactly.
The other tests serve the purpose of determining whether or not the series converges, not to what finite number it converges.
Test Format Results
Alternating
1
1 )1(
n
n
n b , with bn 0 for all n.
If the following two criteria are met, then the series converges:
bn+1 bn for all n N, (i.e., bn is a decreasing sequence.)
n
n b
lim = 0 as a sequence.
Ratio Constuct the ratio
n
n
a
a 1
If
n
n
n a
a 1
lim
< 1 as a sequence, then the series converges.
If
n
n
n a
a 1
lim
> 1 or diverges as a sequence, then the series diverges.
If
n
n
n a
a 1
lim
= 1 as a sequence, then the test fails.
Root Construct the quantity n na
If n n
n a
lim < 1 as a sequence, then the series converges.
If n n
n a
lim > 1 or diverges as a sequence, then the series diverges.
If n n
n a
lim = 1 as a sequence, then the test fails.
