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

Tips for Using Convergence Tests in Math: Geometric, p-Series, Integral, Comparison, Limit, Lecture notes of Mathematics

Tips and instructions on how to use various tests to determine the convergence or divergence of mathematical series. Topics covered include the geometric series test, p-series test, integral test, comparison test, limit comparison test, divergence test, and alternating series test. Each test is explained in detail, including how to apply it and when it is most effective.

Typology: Lecture notes

2021/2022

Uploaded on 09/12/2022

pumpedup
pumpedup 🇺🇸

4.2

(6)

224 documents

1 / 4

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
MATH 21-123
Tips on Using Tests of Convergence
1. Geometric Series Test (GST): The use of this test is straightforward. You only
use this when the series is in the form P
n=0 arn. We say that the series converges if
and only if |r|<1 and the sum is given by, P
n=0 arn=a
1r,where ais the first term
and ris the common ratio. Also note that Pk
n=0 arn=a(1rk+1)
1r.To understand this
formula, look for its proof into your notes. Note that terms of the series could even be
negative.
2. p-Series Test: The series P
n=1
1
npconverges if and only if p > 1. This test can be
proved using integral test(which is described next). You will probably never get to
use this test individually in a problem but nevertheless this test has important role to
play as it supports the working of other tests. We generally use this test along with
comparison and limit comparison test.
3. Integral Test: This test states that if an=f(n) for some function that is nonnegative
and a continuous decreasing (eventually decreasing) function on [1,] then P
n=1 an
converges if and only if R
1f(x)dx converges. Make sure to check all the properties
before you apply this test. In general, you use this test when the terms have the
configuration of a derivative. For example: Pln n
n,P1
nln np,Pnenand etc. The
convergence or the divergence for some of the above mentioned examples may also be
shown using other tests, for instance, we could use comparison test for the first one,
Root test or the Ratio test for the third one.
4. Comparison Test (CT): First of all, note that this test is only valid for positive term
series (like the integral test). This test states that if Panis a series with positive
terms and
*anbnsuch that Pbnconverges then Panalso converges.
*anbnsuch that Pbndiverges then Panalso diverges.
Typically, you choose bnto be either the p-series or the geometric series(basically the
series about which you know quite a lot in terms of its convergence). Please be
warned about the right inequalities. It is easy to fall into the trap of justifying
the wrong inequality.
5. Limit Comparison Test (LCT): This test is also valid for positive term series
(Pan, an0) only. Again you choose bnas in the case of the comparison test. The
only difference is(which makes this test easy to use) that instead of checking for in-
equalities, you look for the limit of an
bn.
Formally speaking, this states that if Panis a series with positive terms and you
choose Pbnsuch that limn→∞
an
bnis non-zero and finite then Panconverges(diverges)
1
pf3
pf4

Partial preview of the text

Download Tips for Using Convergence Tests in Math: Geometric, p-Series, Integral, Comparison, Limit and more Lecture notes Mathematics in PDF only on Docsity!

MATH 21-

Tips on Using Tests of Convergence

  1. Geometric Series Test (GST): The use of this test is straightforward. You only use this when the series is in the form

n=0 ar

n. We say that the series converges if and only if |r| < 1 and the sum is given by,

n=0 ar

n (^) = a 1 −r ,^ where^ a^ is the first term and r is the common ratio. Also note that

∑k n=0 ar

n (^) = a(1−rk+1) 1 −r.^ To understand this formula, look for its proof into your notes. Note that terms of the series could even be negative.

  1. p-Series Test: The series

n=

1 np^ converges if and only if^ p >^ 1. This test can be proved using integral test(which is described next). You will probably never get to use this test individually in a problem but nevertheless this test has important role to play as it supports the working of other tests. We generally use this test along with comparison and limit comparison test.

  1. Integral Test: This test states that if an = f (n) for some function that is nonnegative and a continuous decreasing (eventually decreasing) function on [1, ∞] then

n=1 an converges if and only if

1 f^ (x)^ dx^ converges. Make sure to check all the properties before you apply this test. In general, you use this test when the terms have the configuration of a derivative. For example:

∑ (^) ln n n ,^

nln np^ ,^

ne−n^ and etc. The convergence or the divergence for some of the above mentioned examples may also be shown using other tests, for instance, we could use comparison test for the first one, Root test or the Ratio test for the third one.

  1. Comparison Test (CT): First of all, note that this test is only valid for positive term series (like the integral test). This test states that if

an is a series with positive terms and

  • an ≤ bn such that

bn converges then

an also converges.

  • an ≥ bn such that

bn diverges then

an also diverges.

Typically, you choose bn to be either the p-series or the geometric series(basically the series about which you know quite a lot in terms of its convergence). Please be warned about the right inequalities. It is easy to fall into the trap of justifying the wrong inequality.

  1. Limit Comparison Test (LCT): This test is also valid for positive term series (

an, an ≥ 0) only. Again you choose bn as in the case of the comparison test. The only difference is(which makes this test easy to use) that instead of checking for in- equalities, you look for the limit of a bnn.

Formally speaking, this states that if

an is a series with positive terms and you choose

bn such that limn→∞ a bnn is non-zero and finite then

an converges(diverges)

if and only if

bn converges(diverges).

Typically, you use this test when you are a given a series with nth^ term given by a rational function in term of n. This by no means imply that you cannot use this test for series other the ones that are given by rational functions, for example, this test works best for

sin (^) nπ 2. Keep in mind that this test fails if the limit turns out to be zero or infinity. In this case, you should look for other test. It is possible that the same series might converge or diverge using some other test.

  1. Divergence test (DT): This test states that if limn→∞ an either does not converge to zero or does not exist then the series

an must diverge. The converse of this is not true. Other than checking the divergence of the series, this test also allows us find the limit of sequences. If the series

an converges then limn→∞ an = 0. This is quite useful. For instance, if you recall we found limn→∞ (^) nnn! = 0 using squeeze theorem. Another way to see this, show that the series

∑ (^) n! nn^ converges which can be done using ratio test.

  1. Alternating Series Test (AST): This test is used for checking the convergence of an alternating series. As opposed to the divergence test, this test allows us to claim the convergence of the series

(−1)nan if in addition to lim an = 0 we have that all the terms an are positive and decreasing.

Formally speaking, if

(−1)nan is a series with an ≥ 0 for all n such that

  • an is a decreasing sequence(eventually decreasing).
  • limn→ 0 an = 0

then

(−1)nan converges.

The converse of the above is not true. In other words, the above test is inconclusive if an’s fail to decrease. Note that if lim an 6 = 0 then lim(−1)nan does not exist which implies that

(−1)nan fail to diverge by divergence test.

  1. Ratio Test: This test is very useful in checking for the absolute convergence of the series. Keep in mind that the absolute convergence for positive term series is same as convergence which means that this can be used for any kind of series.

This test states that given any series

an.

  • If limn→∞ |an a+1n | < 1 then

an converges absolutely.

  • If limn→∞ |an a+1n | > 1 then the given series

an diverges.

  • If limn→∞ |an a+1n | = 1 then nothing can be said about the series. In other words, we say that the ratio is inconclusive.

(b) Determine whether the series

n→∞(−1) n(√n + 1 + √n)n (^) converges or diverges.

We do not NEED Root test for this one, though, we can use if we want. Note that limn→∞(

n + 1+

n)n^ == ∞)∞^ = ∞. This implies that limn→∞(−1)n(

√ n^ + 1+ n)n^ does not exist. Hence the given series

n→∞(−1)

n(√n + 1 + √n)n (^) diverges by divergence test.

If the terms are rational functions of n then this test is very difficult to apply and also fails almost every time. Even for simple example such as

n this test fails. Indeed if an = (^1) n and we consider

lim n→∞ |an|^1 /n^ = lim n→∞

n

1 /n = lim n→∞

n^1 /n^

using the fact that limn→∞ n^1 /n^ = 1. This implies that the root test is inconclusive.

Summary of Some Important Tips

  1. Rational terms are best handled with comparison or limit comparison test with p-series test. NO Root or Ratio test for rational functions of n.
  2. Powers of n — Root test.
  3. Factorials and the combination of factorials and powers — Ratio test.
  4. Divergence test is inconclusive if limn→∞ an = 0.
  5. CT is inconclusive if (your series) ≤ (divergent series) or (you series) ≥ (convergent series) or if any of the terms in either sequence are negative.
  6. LCT is inconclusive if limn→∞ a bnn = 0 or ∞, or if any of the terms in either sequence are negative.
  7. AST is inconclusive if an is not decreasing or positive.
  8. The Ratio test if inconclusive if limn→∞ |an a+1n | = 1.
  9. The Ratio test if inconclusive if limn→∞ |an|^1 /n^ = 1.
  10. The tests which are only valid for positive terms can be used for the series with negative terms as well by taking the absolute value of the terms. This can help us determining at least the absolute convergence of the series.

Last Remark: Absolute convergence means everything converges, that is,

|an| converges which further implies that

an convergence. Thus it can thought of as the “strong convergence”. On the other hand, conditional convergence means that the series

an converges BUT

|an| diverges. Thus we can think of this convergence as “ weak convergence” or “partial convergence”.