Series Review. a i converges if lim. i=1. a i. lim S n = lim i=1. 2 k(k + 2) converges. k=1. k=1

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1 Defiitio: We say that the series S = Series Review i= a i is the sum of the first terms. i= a i coverges if lim S exists ad is fiite, where The above is the defiitio of covergece for series. order to see if a series coverges, we eed to fid This meas that i practice, i lim S = lim Example: Determie whether or ot 2 kk 2) coverges. k= First use partial fractio decompositio to rewrite 2 kk 2) as k ) k 2 k= k= To check for covergece, we first eed to fid a formula for the th partial sum, S. S = S 2 = S = S = ) ) ) ) ) ) 4 ) 5 2 ) 4 ) 5 i= a i 4 ) 6 We keep fidig partial sums util we otice a patter that allows us to fid a geeral formula for S : S = 2 2 Now we are able to check for covergece: lim S = lim 2 ) = 2 2 = 2 Because the limit of the th partial sums exists ad is fiite, 2 k= kk 2) coverges.

2 The series i the previous example is telescopig. Telescopig series are ice due to the fact that we ca fid a closed-form expressio for S, i.e. we ca write S i a very simple way that shows its depedece upo. But for may series, we are uable to do this, ad as a result, usig the defiitio of covergece directly becomes very difficult. I that case, we resort to the Covergece/Divergece Tests discussed below. While we ca use these to check for covergece/divergece the defiitio of covergece is what was stated above.. Test for Divergece If lim a 0, the a must diverge. Note: The opposite is ot ecessarily true. diverge. If lim a = 0, The cotrapositive of this statemet is also true: If This does ot mea the followig: If lim a = 0 the a may coverge or a coverges the lim a = 0. a coverges. ) Always thik of the harmoic series as your couterexample to make k k= sure you use this i the correct directio - its terms go to zero, but the series diverges. 2. Geometric Series A geometric series is a series of the form ar i. This series coverges if r < ad diverges if r. A fiite geometric series always coverges, ad we have ar i = a r ) r For a coverget geometric series, we ca fid the sum exactly: i=0 i=0 ar i = i=0 a r Eve though we have this ice cocise formula to use i fidig the sum of a geometric series, we derived it usig the defiitio of covergece. We first foud the formula for the th partial sum the fiite geometric series above) ad the cosidered what happeed as was set to ifiity. 2

3 . Itegral Test Cosider the series is: a. If f) = a for every iteger, ad for < x <, fx) positive cotiuous decreasig The: a) If b) If fx)dx coverges, the fx)dx diverges, the a coverges. a diverges. To see where this statemet comes from, we ca look at a represetative fx) o the iterval [0,8]. From the images above, we see that 8 a i 8 7 fx) dx a i i=2 i= This exteds to ay iterval [,N], ad eve to [, ]. As a result of the Itegral Test, we ca say that fx) dx ad a i either both coverge, or both diverge. i=

4 is called a p-series ad we ca use the Itegral Test to show that it coverges if p > ad diverges if p. p Error Estimates with the Itegral Test: If we approximate a i by its th partial sum, the error is i= a i i= a i = i= i= a i If we represet this sum i two differet ways, we ca fid both a upper boud represetig it as a right had sum) ad a lower boud represetig it as a left had sum). Usig the above images as a guide, we get the error bouds: fx) dx Error) fx) dx 4. Compariso Test Suppose a ad b are series with positive terms. The: a) If b is coverget ad a b for all, the a is coverget. b) If b is diverget ad a b for all, the a is diverget. As a result, whe you re usig the Compariso Test, you wat to try to boud your series above by a series that coverges, or below by a series that diverges. 4

5 The Compariso Test is very useful whe you have a series that looks like a p-series, or looks like a geometric series. I that situatio, you ca compare the give series to the p-series or geometric series that it resemebles. Sometimes, whe you re tryig to use the Compariso Test, you will oly be able to boud your series above by a series that diverges, or below by a series that coverges, i which case the Compariso Test gives us o iformatio. I this case, you may wat to use the Limit Compariso Test. 5. Limit Compariso Test Suppose a ad a b are series with positive terms. If lim = c, where 0 < b c <, the either both series coverge, or both series diverge. Whe usig the Limit Compariso Test, the other series is the oe that failed to give ay iformatio with the Compariso Test. Why do we eed 0 < c < i orer to say that the series behave the same? If a = 2 diverges. ad b = a, the lim = 0. Here, a coverges ad b b a b =. Here, a diverges ad b co- If a = ad b =, the lim 2 verges. 6. Alteratig Series Test Suppose a alteratig series ) a satisfies both of the followig coditios: lim a = 0 a < a for all after some fixed value The: ) a coverges If Error) is the error i approximatig Error) < a ) a by the th partial sum, 5

6 Note that you ca t ever say that a series diverges by the Alteratig Series Test. Istead, if oe of the coditios does ot hold, we have to use aother test ofte the Test for Divergece). Examples: Whe the coditios do t hold ) 2 This alteratig series does ot satisfy the first coditio of the AST. it diverges by the Test for Divergece This series does ot satisfy the secod coditio of the AST the terms are ot decreasig). By rewritig the 2 sum as, we ca see that the series diverges. 2 Examples: Just because a series icludes ) k, does t mea that you must or ca) use AST! ) k 2k k k= This series is *ot* alteratig, so you caot use the Alteratig Series Test. Istead, this series diverges by the Test for Divergece. ) k k= 2 k This series *is* alteratig, ad you *ca* use the Alteratig Series Test. But, also, ote that this is a geometric series with r = <, so we ca use that to 2 quickly see that the series coverges. Also, the AST does ot allow us to fid the sum of the series, but by usig our kowledge of geometric sereis, we ca fid its sum. 7. Absolute Covergece Theorem If a coverges, the a coverges. Note: The opposite is NOT always true. The covergece of a tells us othig about the covergece or divergece of a 6

7 Defiitio: We say a series a is absolutely coverget if a coverges. We say a series b is coditioally coverget if b coveres, but b diverges. Examples: ) absolutely coverget: ) 2) coditioally coverget: 8. Ratio Test a Let lim = L. The: a a) If L <, b) If L >, c) If L =, test). 2 ) a coverges. a diverges. a i.e. the Ratio Test is icoclusive so you have to use aother If you fid that a series coverges usig the Ratio Test, it coverges absolutely. More o Series A fiite sum of fiite terms) always coverges. a coverges =0 a coverges for some 0 < c <. =c If a series diverges, the error i approximatig it by partial sum is ifiity ad that s why we do t care to estimate the error i this case). We ca say that a b = a b ) oly if both But thigs chage whe oe of these series or both) diverges. a ad b coverge. 7

8 Example: If a = ad b =, both a ad But a b ) = We ca t assume 0) which coverges. a b = a b ) Example: If a = = b, the both But a b ) = a ad b diverge. b diverge. which coverges, so these cat be equal. 2 8