Ap Calculus Exam How Do You Know What Units to Use

Contents

1. 1 Welcome
2. 2 Infinite Sequences
3. iii Introduction to Infinite Series
4. iv Serial Tests for Convergence and Divergence
   1. 4.one Nth Term Test for Difference
   2. 4.2 Geometric Serial Exam
   3. four.3 Harmonic Series
   4. four.4 P-Serial Test
   5. 4.5 Integral Examination
   6. four.six Comparison Tests
      1. 4.6.one Limit Comparison Test
      2. 4.6.2 Straight Comparison Test
   7. 4.seven Alternating Series Test
   8. 4.8 Ratio Exam
   9. 4.nine Root Test
5. 5 Error

Welcome

Welcome to our AP Calculus Series Tests for Convergence wiki! Hither we have posted the essential convergence tests that yous need to know for your AP Calculus BC examination. Nosotros will outline the essential concepts required for yous to successfully take advantage of the post-obit tests and include gratis examples to help solidify your understanding. A basic knowledge of algebra, limits, and integration is essential in order to comprehend this complex topic.

Before nosotros begin our deviation into the earth of series, convergent and divergent akin, we must beginning found command of the cardinal concepts involved in such series.  We shall begin with an introduction of the sequence and hash out some of its basic principles before we go forrard.Here we go!

Space Sequences

A sequence is just a list of numbers written in a definite order like then:

a₁, a₂, a_iii, a₄ .... a_north ....

The Sequence has a first, second, tertiary, fourth, and nth term.

When it comes to infinite sequences, they tin take the characteristic of either convergence or difference depending on their behavior as the terms go on infinitely.

We tin can see by the post-obit that we take a sequence whose terms converge upon a particular value L as we approach a _∞.

We can say that a sequence is convergent if  = L  and diverges if =∞ .

Another characteristic of sequences is monotonicity.  A sequence is monotonic if it is either increasing so that a _n < a _n+ane OR  decreasing so that a _{n >} a _{due north+i}  for all n > one .

With these principles in mind, we can now begin to delve into Series.

Introduction to Infinite Series

Serial are related to sequences in that series are sums of sequence terms. This is seen by the relation: = S_northward

where S_nrepresents the nth partial sum of the series.  The partial sum of a series denoted by S _n is simply the sum of the starting time n terms of the sequence a_northward.For example: Southward _two  = a ₁ + a ₂ .

We can see that sinceinfinitely many terms volition be added together in a series, it will be far more difficult for series to converge than their sequence counterparts.  As nosotros will come to learn, for series to converge they not simply need to accept the sum of Southward _∞ equal a finite number only their sequences a _n MUST converge to 0 so that their terms subtract quickly enough.

Series Tests for Convergence and Departure

Knowing the basics behind sequences and series, now nosotros tin can begin to explore the more circuitous arena of testing for the convergence or difference of a particular series using the different methods we shall address in this section.  With the post-obit methods nosotros will be able to evaluate almost any series for its convergence.  Let'south begin!

Nth Term Test for Departure

To make up one's mind if a series diverges or converges , we can use the Nth term test for deviation by finding .

• If ≠ 0 or if the limit does not exist, and then the series diverges.

• If = 0, then the series may or may non converge. In this case, the test is inconclusive and we must use some other examination.

This test tin employ to any series and should be the first test used in determining the convergence or deviation of a series.  If you find a series divergent by this method, yous need non continue testing!  If the series converges, y'all must go on to ane of the other tests nosotros will discuss.

Geometric Series Exam

Geometric series are a special type of series such that each term is multiplied by a common ratio, r. The value, a, represents the first term in the serial; thus the series is represented by:

A geometric series converges if |r|<1, and diverges if |r| > 1. If the geometric series is convergent, then the sum tin be found by:

    =South _n

You should use this test if yous notice all the terms being multiplied past a mutual ratio such every bit ane/2.

Examples:

Harmonic Serial

The harmonic series is a divergent serial such that

The harmonic series is useful when using either the Direct or Limit Comparing Tests because it serves every bit a reference series by which to compare unknown series to.  To employ the harmonic series successfully in these other tests, it is important to exist able to recognize it on sight.  Practise non find yourself testing some uncomplicated, abiding, multiplied variation of this serial: you should already know information technology diverges!

P-Series Test

The p-series test is used to determine the convergence of an infinite series of the form:

Where p is any positive, real number.This test tells us that the series converges when p>1. When p< 1, the series diverges.

The p-series is also useful when using either the Direct or Limit Comparison Tests but similar the harmonic series.

Examples:

Integral Test

---

For an space series, , if the function, f(n) = a_n, on the interval (one, ∞] is:

1.  continuous
2.  positive
3.  decreasing

so the series  either diverges or converges with the integral .

Unfortunately, it isnecessary to prove the higher up mentioned atmospheric condition earlier performing the Integral Test considering we cannot be sure that the improper integral of f(10) diverges or converges in the same manner equally unless we can verify that f(x) and a_n comport identically.  Because of the extra preparatory work and the potentially difficult evaluation of improper integrals involved, the Integral Test is 1 of the least friendly tests to employ especially if another method presents itself as easier or quicker.

For example:

This would suggest that the respective serial would diverge as the in a higher place

integral does (granted that the weather have been proven beforehand).

Comparing Tests

The comparison test come in ii singled-out flavors: limit and direct comparison.

Limit Comparison Examination

If where L is a finite number and Fifty > 0, then the two series converge or diverge together.

Example:

This test tin can be useful with series that do not arrange to easy evaluation with previous tests.  Mostly, the ideal series for this examination would exist one whose terms are represented by a quotient of long polynomials or other like serial.  Every bit you may guess, this would make for a grueling manipulation if tested using the Direct Comparing Test ; instead we proceed with the Limit Comparison Test by finding an appropriate refe rence serial (the terms of which are represented b_{due north} ) for which the convergence/divergence is known and evaluate.

Straight Comparing Test

a. If is convergent and a_n ≤ b_north for all n, then  is also convergent.

b. If  is divergent and a _n ≥ b _north for all due north,  and then is also divergent.

Example:

We normally utilize this test when a relationship tin be easily established by way of inequality to show that the unknown series' terms are always greater than those of a known divergent series or otherwise always smaller than those of a known convergent series.  If this is not easily done due to difficulty with algebraic manipulation or elsewise, the Limit Comparison Test or some other test may be a better option.

Alternate Series Test

Given an alternating series ,  the alternating series will converge if the following two weather condition are met:

i. a_n ≥ a_n+1 > 0

ii. = 0

If either status is not met, then the series diverges.

Case (though not very thorough on demonstrating proof of the weather):

Nosotros must note in our give-and-take about alternating serial the distinction between absolute and conditional convergence. When it comes to alternating serial (series that have a cistron of (-1)^northward-1 causing the terms to alternate betwixt positive and negative), we tin can classify convergence into absolutely convergent and conditionally convergent based upon the behavior of the terms a_north.  If we can prove that the alternating series converges , every bit we exercise with the alternating serial examination, and that it does not converge without the alternating factor, (-i)^n-i , past some other test, nosotros can conclude that the series converges conditionally.  On the other hand, if nosotros use the Alternating Series Test and notice that a series converges conditionally but then decide independently that the series with | a_n | also converges, we can conclude that the series converges absolutely.  In the post-obit tests where the limit of an absolute value ratio is taken, a result indicating convergence actually indicates that the tested series is absolutely convergent.

Ratio Test

The ratio exam is useful for determining the convergence of a wide variety of series, particularly those which comprise factorials.

For an infinite series, , you perform this test by obtaining the .

• If R < 1, and then the series converges absolutely
• If R > 1, so the series diverges
• If R = one, and then the test is inconclusive

Example:

As mentioned before, this test can make up one's mind whether a serial is divergent or absolutely convergent based upon the value of the ratio R.  In the instance that R = 1, you must use another series test to conclude with certainty the condition of the series' divergence/convergence.  If R does equal 1, practise not apply the Root Test instead because it will yield an equivalent inconclusive result of L = 1!

Root Exam

For an infinite series, , you perform the root test by finding the .

• If L < one, then the series converges absolutely
• If L > 1, so the series diverges
• If 50 = 1, then the test information technology inconclusive

Example:

Generally nosotros use the Ratio Test to make up one's mind the divergence/convergence of series containing factorials, exponents, and other more than complex terms. We use The Root Test nether the circumstances that the unabridged quantity a_n is raised to a ability of northward to eliminate the power and evaluate the limit of the isolated a_n . Considering the Ratio test is more than widely applicable to diverse serial, we use it more frequently than the more specialized Root Examination.

Error

Regarding Alternating Series, nosotros tin determine the error of the nth partial sum S_{due north} in approximating sum of the serial .  The error denoted by R_{due north} is always less than the accented value of the next term a_{due north+ane} . Thus we have:

R _n ≤ | a _n+1|.

Example:

Thank you for visiting our Series Tests Wiki Folio.  We promise nosotros take helped you lot learn how to finer tackle series and or clarified any questions you may have had!

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