Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Convergence Tests for Infinite Series in Mathematics, Study Guides, Projects, Research of Mathematics

An overview of various tests used to determine the convergence or divergence of infinite series in mathematics. Topics include the ratio test, root test, alternating series test, integral test, n-th term test, geometric series test, comparison test, and limit comparison test. Each test is explained with its corresponding conditions for convergence or divergence.

Typology: Study Guides, Projects, Research

2021/2022

Uploaded on 09/12/2022

ekadant
ekadant ๐Ÿ‡บ๐Ÿ‡ธ

4.3

(31)

268 documents

1 / 2

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Math 142 Series
Mnemonic: PARTING CC
P-series, Alternating Series Test, Ratio and Root Tests, Telescoping Series, Integral Test, N-th Term Test, Geometric
Series Test
Comparison Test, Limit Comparison Test
p-Series
Form:
โˆž
X
n=1
1
np
Converges: If p > 1.
Diverges: If pโ‰ค1.
Remember to look for some constant multiple of this series also. (Like an=2
n5)
Alternating Series Test
Form:
โˆž
X
n=1
(โˆ’1)nunor
โˆž
X
n=1
(โˆ’1)n+1un
Converges: If (a) un+1 โ‰คunfor all nand (b) lim
nโ†’โˆž
un= 0.
This test only applies to alternating series. Also, note that we only need that {un}is eventually a decreasing
sequence since a finite number of terms does not affect convergence.
Ratio Test
Suppose we have the series Pan. Define L= lim
nโ†’โˆž ๎˜Œ
๎˜Œ
๎˜Œ
๎˜Œ
an+1
an๎˜Œ
๎˜Œ
๎˜Œ
๎˜Œ
.
(1) If L < 1 the series is absolutely convergent (and hence convergent).
(2) If L > 1 the series is divergent.
(3) If L= 1 the series may be divergent, conditionally convergent, or absolutely convergent.
Use this test when the series contains factorials or other products (including a constant raised to the
nth power).
Root Test
Suppose we have the series Pan. Define L= lim
nโ†’โˆž
n
p|an|= lim
nโ†’โˆž |an|1/n.
(1) If L < 1 the series is absolutely convergent (and hence convergent).
(2) If L > 1 the series is divergent.
(3) If L= 1 the series may be divergent, conditionally convergent, or absolutely convergent.
Try this test when an= (bn)n.
Telescoping Series
A telescoping series is a series whose partial sums SNeventually only have a fixed number of terms after
cancellation. You can then evaluate the limit lim
Nโ†’โˆž
SNto determine whether the series converges or diverges.
(Partial fraction decomposition may be necessary.)
1
pf2

Partial preview of the text

Download Convergence Tests for Infinite Series in Mathematics and more Study Guides, Projects, Research Mathematics in PDF only on Docsity!

Math 142 Series

Mnemonic: PARTING CC

P-series, Alternating Series Test, Ratio and Root Tests, Telescoping Series, Integral Test, N-th Term Test, Geometric Series Test Comparison Test, Limit Comparison Test

p-Series

Form:

โˆ‘^ โˆž

n=

np

Converges: If p > 1. Diverges: If p โ‰ค 1.

Remember to look for some constant multiple of this series also. (Like an = (^) n^25 )

Alternating Series Test

Form:

โˆ‘^ โˆž

n=

(โˆ’1)nun or

โˆ‘^ โˆž

n=

(โˆ’1)n+1un

Converges: If (a) un+1 โ‰ค un for all n and (b) lim nโ†’โˆž un = 0.

This test only applies to alternating series. Also, note that we only need that {un} is eventually a decreasing sequence since a finite number of terms does not affect convergence.

Ratio Test

Suppose we have the series

an. Define L = lim nโ†’โˆž

an+ an

(1) If L < 1 the series is absolutely convergent (and hence convergent). (2) If L > 1 the series is divergent. (3) If L = 1 the series may be divergent, conditionally convergent, or absolutely convergent.

Use this test when the series contains factorials or other products (including a constant raised to the nth power).

Root Test

Suppose we have the series

an. Define L = (^) nlimโ†’โˆž^ n

|an| = (^) nlimโ†’โˆž |an|^1 /n.

(1) If L < 1 the series is absolutely convergent (and hence convergent). (2) If L > 1 the series is divergent. (3) If L = 1 the series may be divergent, conditionally convergent, or absolutely convergent.

Try this test when an = (bn)n.

Telescoping Series

A telescoping series is a series whose partial sums SN eventually only have a fixed number of terms after cancellation. You can then evaluate the limit lim N โ†’โˆž SN to determine whether the series converges or diverges.

(Partial fraction decomposition may be necessary.)

1

2

Integral Test

Suppose that f (x) is a continuous, positive, decreasing function on the interval [k, โˆž) and that f (n) = an then

(1) If

k

f (x) dx is convergent so is

โˆ‘^ โˆž

n=k

an.

(2) If

k

f (x) dx is divergent so is

โˆ‘^ โˆž

n=k

an.

This test only applies to series that have positive terms. Try this test when f (x) is easy to integrate.

N -th Term Test for Divergence

If lim nโ†’โˆž an 6 = 0 then

an diverges.

Look for the same degree in the numerator and denominator. Remember: If lim nโ†’โˆž an = 0 then we know nothing.

Geometric Series

Form:

โˆ‘^ โˆž

n=

arnโˆ’^1

Converges: If |r| < 1. Diverges: If |r| โ‰ฅ 1.

Some algebraic manipulation is often required to get a geometric series into the correct form. (Remember r is the common ratio: r = an a+1n .)

Comparison Test

Suppose that

an and

bn are series with positive terms:

(1) If

bn is convergent and an โ‰ค bn for all n, then

an is also convergent. (2) If

bn is divergent and an โ‰ฅ bn for all n, then

an is also divergent.

Use this test if the given series is similar to a p-series or geometric series. Remember, the terms of the series being tested must be smaller than a convergent series or larger than a divergent series. This test is often a last resort; other tests are often easier to apply.

Limit Comparison Test

Suppose that

an and

bn are series with positive terms:

If lim nโ†’โˆž

an bn

= C, where C > 0 is a finite number, then either both series converge or both series diverge.

Try this test if an is a rational expression involving only polynomials or polynomials under radicals. This test is easier to apply than the Comparison Test.