SUMMARY OF SEQUENCES AND SERIES

SUMMARY OF SEQUENCES AND SERIES Importat Defiitios, Results ad Theorems for Sequeces ad Series Defiitio. A sequece {a } has a limit L ad we write lim a = L if for every ɛ > 0, there is a correspodig iteger N such that if > N the a L < ɛ. Theorem. If lim x f(x) = L ad f() = a, whe is a iteger, the lim a = L. Theorem 2. If lim a = 0, the lim a = 0. Result. The sequece {r } is coverget if < r ad diverget for all other values of r. 0 if < r < lim r = if r = Theorem 3. Every bouded, mootoic sequece is coverget Defiitio. Give a series a = a + a 2 + + a +, let s deote the fiite its th partial sum s = If the sequece {s } is coverget ad lim s = s exists as a real umber, the the series a is called coverget ad we write a a = s The umber s is called the sum of the series. If the sequece {s } is diverget, the the series is called diverget. Result (Geometric Series). The geometric series ar = a + ar + ar 2 +

2 SUMMARY OF SEQUENCES AND SERIES is coverget if r < ad its sum is ar = a r If r, the the geometric series is diverget. r < Theorem 4. If the series a is coverget, the lim a = 0. Theorem 5 (The Test for Divergece). If lim a does ot exist of if lim a 0, the the series a is diverget. Result. If a ad b are coverget series, the so are the series c a, (a + b ) ad (a b ), ( c is a costat) ad () ca = c a (2) (a ± b ) = a ± b Result (The Itegral Test). Suppose f is a cotiuous, positive, decreasig fuctio o [, ) ad let a = f(). The () If f(x) dx is coverget, the a is coverget. (2) If f(x) dx is diverget, the a is diverget. Result (p-series test). The p-series The series is called the Harmoic Series. is coverget if p > ad diverget if p. p Result (The Compariso Test). Suppose that a ad b are series with positive terms. () If b is coverget ad a b for all, the a is also coverget. (2) If b is diverget ad a b for all, the a is also diverget. Result (The Limit Compariso Test). Suppose that a ad b are series with positive terms. If a lim = c b where c is a fiite umber ad c > 0, the either both series coverge or both diverge.

SUMMARY OF SEQUENCES AND SERIES 3 Defiitio. A alteratig series is a series whose terms are alteatively positive ad egative. Result (The Alteratig Series Test). A alteratig series ( ) b = b b 2 + b 3 + b > 0 is coverget if it satisfies the followig coditios () b + b for all (2) lim b = 0. Defiitio (Absolutely Covergece). A series a is called absolutely coverget if the series of the absolute values a is coverget. Defiitio. A series a is called coditioally coverget if it is coverget but ot absolutely coverget. Theorem 6. If a series a is absolutely coverget, the it is coverget. Result (The Ratio Test). Give a series a, () If lim a + a = L <, the the series a is absolutely coverget. (2) If lim a + a = L > or lim a + a =, the the seriest a is diverget. (3) If lim a + a = L =, the ratio Test is icoclusive. Result (The Root Test). Give a series a, () If lim a = L <, the the series a is absolutely coverget. (2) If lim a = L > or lim (3) If lim a = L =, the root Test is icoclusive. a =, the the seriest a is diverget. Defiitio (Power Series). A power series, cetered at a, is of the form c (x a) = c 0 + c (x a) + c 2 (x a) 2 +, where a is a fixed umber, x is a variable. c s are costats, called the coefficiets of the series. Theorem 0.. For a give series c (x a), there are oly three posssibilities:

4 SUMMARY OF SEQUENCES AND SERIES () The series oly coverges for x = a. (2) The series coverges for all values of x. (3) There is a positive umber R such that the seies coverges whe x a < R ad diverges if x a > R. The umber R i (3) is called the radius of covergece of the power series. I (), R = 0, while i (2), R =. Remark. Note that the series may or may ot coverge if x a = R. What happes at these poits will ot chage the radius of covergece. Defiitio. The iterval of all x s, icludig the edpoits if eed be, for which the power series coverges is called the iterval of covergece of the series. For (), the iterval of covergece is {a}. For (2), the iterval of covergece is (, ). For (3), the iterval of covergece is either ( a R, a+r ), ( a R, a+r ], [ a R, a+r ) or [ a R, a+r ], depedig whether the series coverges for the edpoits of the iterval. Theorem 0.2. If f has a power series represetatio at a, that is, if f(x) = c (x a) x a < R the it s coefficiets are give by the formula c = f () (a) I other words, if f has a power series expasio at a, the it must be of the form f(x) = f () (a) (x a) This series is called the Taylor series of fuctio about a (or cetered at a). Defiitio (Maclauri Series). I the special case that a = 0, the series f(x) = c x where c = f () (0), is called the Maclauri series of the fuctio f(x). Importat Maclauri Series The followig Maclauri series are very importat ad you are expected to remember them: () x = + x + x2 + x 3 + = x, < x <. (2) + x = x + x2 x 3 + = ( ) x, < x <.

(3) e x = + x + x2 2! + x3 3! + = SUMMARY OF SEQUENCES AND SERIES 5 (4) cos(x) = x2 2! + x4 4! x6 6! + = x, x i (, ). (5) si(x) = x x3 3! + x5 5! x7 7! + = ( ) x 2 (2)! ( ) x 2+ (2 + )!, x i (, )., x i (, ) Remark. To avoid cofusio betwee the series for si x ad cos x, remember that: si x is a odd fuctio, that is, si( x) = si x, so the series for si x has oly odd expoets. cos x is a eve fuctio, that is cos( x) = cos x, so the series for cos x has oly eve expoets.