Calculus BC ad BCD Drill o Sequeces ad Series!!! By Susa E. Catey Walut Hills H.S. 2006
Sequeces ad Series I m goig to ask you questios about sequeces ad series ad drill you o some thigs that eed to be memorized. It s importat to be fast as time is your eemy o the AP Eam. Whe you thik you kow the aswer, (or if you give up ) click to get to the et slide to see the aswer(s).
What s the differece betwee a sequece ad a series?
Didyagetit?? A sequece is a list (separated by commas). A series adds the umbers i the list together. Eample: Sequece: 1, 2, 3, 4,,, Series: 1 + 2 + 3 + 4 + + + (ote that i calculus we oly eamie ifiite sequeces ad series)
What symbol(s) do we use For a sequece? For a series?
OK so far?? { } a represets a sequece a represets a series
How do you fid the limit of a sequece? a 1, a 2, a 3, a 4, a 5, a 6, a 7, a 8, a 9, a 100, a 101, a 1000, a 1001, a 1002, a 7000000, a 7000001, a, where s it goig?
Simple! Just take the limit as This image caot curretly be displayed. Remember, you ca treat as tho it were a (You may have to use L Hopital s Rule) This image caot curretly be displayed.
OK that s about it for sequeces. Let s move o to series. There are 2 special series that we ca actually fid the sum of What are their ames?
Geometric ad Telescopig What does a geometric series look like? How do you fid it s sum? Why is it called geometric?
Geometric series are of the form: a ( r ) A geometric series oly coverges if r is betwee -1 ad 1 The sum of a coverget geometric series is: the first term 1 r See the et slide for a possible aswer as to why these series are called geometric
Maybe this is why the ame geometric sice the idea origiated from a physical problem The aciet Greek philosopher Zeo (5th cetury BC) was famous for creatig paradoes to ve the itellectuals of his time. I oe of those paradoes, he says that if you are 1 meter away from a wall, you ca ever reach the wall by walkig toward it. This is because first you have to traverse half the distace, or 1/2 meter, the half the remaiig distace, or 1/4 meter, the half agai, or 1/8 meter, ad so o. You ca ever reach the wall because there is always some small fiite distace left. The theory of ifiite geometric series ca be used to aswer this parado. Zeo is actually sayig that we caot get to the wall because the total distace we must travel is 1/2 + 1/4 + 1/8 + 1/16 +..., a ifiite sum. But this is just a ifiite geometric series with first term ½ ad commo ratio ½, ad its sum is (½)/(1 - ½)=1. So the ifiite sum is oe meter ad we ca ideed get to the wall.
What about telescopig (or collapsig ) series? What are telescopig series? What types of series do you suspect of beig telescopig ad how do you fid their sum?
If whe epaded, all the terms i the middle cacel out ad you are left with oly the first term(s) because the th term heads to zero, the the series is telescopig or collapsig Suspicious forms: 1 1 ± a+ b c+ d or 1 ( a+ b)( c+ d) The latter ca be separated ito 2 fractios ad the observed. Always write out the first few terms as well as the last th terms i order to observe the cacellig patter. Also! Make sure that the o cacellig th term goes to zero. Telescopig series ca be cleverly disguised! So be o the look out for them.
I geeral, to fid S, the sum of a series, you eed to take the limit of the partial sums: S What s a partial Sum?
You sum some of the sum Ha ha sum some of the sum I kill myself! S = a + a + a +... + 1 2 3 a
I other words: a = lim S = S (If S eists)
If a 0 What does that tell you about the series?
The series diverges. Help!!
What if a 0?
The the series might coverge. That s why we eed all those aoyig #$@%^&*($#@* tests for covergece (comig up) which are so difficult to keep straight Why if I had a dollar for every studet who ever thought that if the a s wet to zero that meat the series coverged, I d be istead of
Alteratig Series Test What does it say? Warig this picture is totally irrelevat.
If the terms of a series alterate positive ad egative AND also go to zero, the series will coverge. Ofte there will be (-1) i the formula but check it out ad make sure the terms reeeeeally alterate. Do t be tricked! Also ote that if the series alterates, ad if you stop addig at a, your error will be less tha the et term: a +1
OK here s a couple of famous series that come i hady quite ofte. What are p-series ad What is the harmoic series?
The harmoic series: diverges most people are surprised! 1 p-series: coverges for p > 1 diverges for p < 1 1 p
What s the itegral test ad Whe should you use it?
The itegral test says that if f ( ) d = c K where K is a positive real umber, the the series coverges also. but NOT to the same umber! f ( ) d (you ca however use to approimate the error for S if is large) If the itegral diverges, the so does the series. Use the itegral test oly if chagig to yields a easily itegratable fuctio.
Now we re movig alog!!
Here are three limits you eed to kow as what happes to: 1. 2. ad fially 3. c ( c 1+ )
The aswers are 1, 1, ad e c respectively. Net questio: What is the Root Test ad whe should you use it?
The root test says: that as If a b < 1 the series coverges. If a b > 1 the series diverges But if a b = 1 the test is icoclusive Use the ROOT TEST whe the terms have s i their epoets.
What is the RATIO TEST? Whe should you use it?
The RATIO TEST should be used whe a cotais! or somethig like! such as: 1 3 5... (2 + 1) It says to compare the limit as of to 1 a + 1 A limit < 1 idicates covergece, > 1 idicates divergece If the limit equals 1 the the test is icoclusive. a
WHEW! Tired Yet?? OK just 2 more tests for covergece Compariso Tests: Direct Compariso & Limit Compariso
Direct Compariso What is it? Whe do you use it?
If you ca show that your positive terms are greater tha a kow diverget series 1 (like or a p-series where p < 1) or smaller tha a kow coverget series (like a p-series where p > 1) the you are usig the Direct Compariso Test. Questio: If it is ot easy to compare the series directly, how do you employ the Limit Compariso Test??
Form a ratio with the terms of the series you are testig for covergece ad the terms of a kow series that is similar: a b If the limit of this ratio as is a positive real # the both series do the same thig i.e. both coverge or both diverge If the limit is zero or ifiity the either you are comparig your series to oe that is ot similar eough or you eed a differet test.
What is a Power Series?
A power series is of the form: a So how do you figure out the values of which yield covergece?
Put absolute value aroud the part ad apply either the ratio or the root test. For eample: lim 1 2 = 1 ( lim 2) 2 Now solve for : 1 < 2 < 1 1 < < 3 = 2 < 1 Checkig the edpoits separately, =3 yields the harmoic series (diverget) ad =1 yields the alteratig harmoic series (coverget). Iterval of covergece is [ 1, 3 ), radius of covergece = 1
What is the Biomial Series Formula?
s ) 1 ( + = = 0! 1)) ( 2)...( 1)( ( s s s s Remember that the fractio has the same umber of fractios (or itegers if s is a iteger) i the umerator as the factorial i the deomiator. Also the iterval of covergece is (-1,1) Eample:... ) ( 1 2 3 ) )( )( ( ) ( 1 2 ) )( ( ) ( 1 1 ) (1 3 2 3 4 3 1 3 2 2 2 3 1 3 2 1 2 3 2 2 3 2 + + + + = +
Do you eed to take a break ad come back i a miute? eat some chocolate maybe? or take a little ap? OK maybe some deep breaths will have to do. Here come some epressios you should have memorized the ifiite series for
1 1 =? Where -1 < < 1
1 + + 2 + 3 +... = = 0 Ready?
1 1 = +? Where -1 < < 1
1 + 2 3 +... = ( 1) = 0 Ready?
l( 1+ ) =?
Ack! Never ca remember that oe so I just itegrate the previous oe. 2 d = 1 + + 1+ 1 3... d l(1 + ) = 1 2 2 + 1 3 3... = = 0 ( 1) + 1 + 1 I kow; I kow hag i there!
si =??? cos =??? ta -1 =???
si = 3 3! + 5 5!... = = 0 ( 1) (2 + 1)! 2+ 1 cos = 1 2 2! + 4 4!... = = 0 ( 1) (2)! 2 ta 1 = 3 3 + 5 5... = = 0 ( 1) (2 + 1) 2+ 1 Note the similarities if you kow oe, do you kow the rest?
OK! Almost doe!! Just four more questios!
What is the formula for a Maclaure Series? (Used to approimate a fuctio ear zero)
=0 f ( )! (0) Ok How about the Taylor Series?
= 0 f ( )! ( a) ( a) Used to approimate f() ear a.
What is the LaGrage Remaider Formula for approimatig errors i NON alteratig series?
k k k a k a f f ) (! ) ( ) ( 0 ) ( = = + 1 1) ( ) ( 1)! ( ) ( + + + = a t f R Where t is some umber betwee a ad The we fid the maimum possible value of ) ( 1) ( t f + Give: R to approimate the error (remaider).
Last questio!!! How do you approimate the error (remaider) for a alteratig series?
Ha! I told you earlier i this presetatio. Remember?
The error i a alteratg series is always less tha the et term. R < a + 1
Cogratulatios! You fiished! Bye bye for ow!
Be sure to check out the power poit drills for: Pre-Calc Topics, Derivatives, Itegrals, Miscellaeous Topics, ad other BC/BCD Topics