Ans: a n = 3 + ( 1) n Determine whether the sequence converges or diverges. If it converges, find the limit.

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Agatha McKinney
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1 . Fid a formula for the term a of the give sequece: {, 3, 9, 7, 8 },... As: a = 3 b. { 4, 9, 36, 45 },... As: a = ( ) ( + ) c. {5,, 5,, 5,, 5,,... } As: a = 3 + ( ) +. Determie whether the sequece coverges or diverges. If it coverges, fid the limit. a = As: lim a = b. a = 3 + As: Diverget. c. a = 3+ 5 As: lim a = 0 + d. a = 9 As: lim a = 3 e. a = cos(/) As: lim a = f. {arcta()} As: lim a = π { } l l() As: lim a = h. { cos(π)} As: Diverget.

2 i. a = l( + ) l As: lim a = 0 (l ) j. a = As: lim a = 0 { k., 3,, 4, 3, 5, 4, } 6,... As: lim a = 0 3. Determie if the series is coveget or diverget. If it is coveget, fid its sum As: Diverget. b. ( ) =0 As: Coverges to c. =0 As: Diverget. + 3 d. =0 As: Diverget. e. (cos ) =0 As: Coverges to cos 4. Determie if the series is coverget or diverget. (Itegral/P-Series) 5 As: Coverget. p = 5 b. + 3 As: Diverget. Itegrate by substitutio.

3 c As: Coverget. P-series d As: Diverget. e. 3 + =0 3 with p = 3. Itegrate usig substitutio. + As: Diverget. Itegrate usig substitutio. 3 + f. ( + ) As: Diverget. Itegrate usig partial fractio As: Coverget. Itegrate with partial fractio. h. (l ) = As: Coverget. Itegrate usig substitutio. i. e =3 As: Coverget. Itegrate by parts. 5. Determie whether the series coverges or diverges. (Compariso/Limit Comparisio) 3 4 = As: Diverget. Compare to harmoic series. b. As: Coverget. Compare to

4 c As: Diverget. Compare to d As: Coverget. Compare to e. + si 0 As: Coverget. Compare to f. As: Diverget. Compare to 3 + As: Coverget. Compare to h = ( ) 3 As: Coverget. Compare or limit compare to i As: Diverget. Compare or limit compare to harmoic series. + 8 j As: Coverget. Compare to

5 k. e / As: Diverget. Compare to harmoic series.! l. As: Coverget. Compare to P-series with p = m. +/ As: Diverget. Limit compare to harmoic series. ( ). si As: Diverget. Limit compare to harmoic series. ( ) o. si As: Coverget. Limit Compare to 6. Determie if the alteratig series is absolutely coverget, coditioally coverget, or diverget. { } + As: Diverget. lim a 0 ( ) b. l( + 4) As: Coditioally coverget. Use alteratig series test for covergece of alteratig series. ( ) l( + 4) = diverges by compariso to l( + 4) harmoic series. ( ) c. + As: Diverget. lim a 0 ( ) d. 3 + As: Absolutely Coverget. Compare to P-series p = 3/

6 e. ( ) e / As: Coditioally Coverget. Use alteratig series test for covergece of alteratig series. ( ) e / = e / diverges by compariso to harmoic series. ( ) l f. As: Coditioally Coverget. Use alteratig series test for covergece of alteratig series. ( ) l = l diverges by compariso to harmoic series. si(π/)! As: Absolutely Coverget. Compare to h. ( π ( ) cos ) As: Diverget. lim a 0 ( i. ) 9 As: Diverget. lim a 0 7. Determie if the series is absolutely coverget, coditioally coverget, or diverget. (Ratio, root, or Compariso) ( ) ( ) 5 As: Diverget. lim a 0 ( ) b. 3 As: Absolutely Coverget. Compare to P-series, p = 3! c.! + As: Diverget. lim a 0!

7 d. ( ) 3 + As: Coditially Coverget. Use alteratig series test for covergece of alteratig series. ( ) 3 + = is diverget by compariso to e. ( )! As: Absolutely Coverget. Ratio test. ( ) arcta f. As: Absolutely Coverget. lim arcta() = π. Comparsio to ( ) l( + ) As: Coditially Coverget. Use alteratig series test for covergece of alteratig series. ( ) l( + ) = diverges by compariso to l( + ) harmoic series. ( ) h. (l ) = As: Absolutely Coverget. Compare to 8. Determie if the give series is coverget or diverget. If it is a alteratig series, determie if it is absolutely coverget, coditioally coverget, or diverget. (Ope) ( + ) As: Coverget. Ratio test. ( ) b. + As: Coditially Coverget. Use alteratig series test for covergece of al- = π/

8 teratig series. series. c. 3 5 =3 ( ) + = + As: Diverget. Compare to harmoic series.! d. ( + )! As: Diverget. Ratio or divergece test. 5 e. e As: Coverget. Ratio test. + f. 3 + As: Diverget. Compariso to harmoic series. ( ) (l ) 3 = diverges by compariso to harmoic As: Coditially Coverget. Use alteratig series test for covergece of alteratig series. ( ) = (l ) 3 = 3 diverges by compariso to harmoic = (l ) series. + h As: Coverget. Compariso to i. si(/) As: Diverget. Divergece test. ( ) j. + 3 As: Coditially Coverget. Use alteratig series test for covergece of alteratig series. ( ) + = + diverges by compariso to.

9 k. (l ) l As: Coverget. Compariso to l. ( ) 3 (e ) l As: Diverget. Limit compariso to harmoic series.! m. e As: Coverget. Ratio test. ( ). ta As: Diverget. Limit compariso to harmoic series. 5 o As: Diverget. Divergece test. l p. ( + ) 3 As: Coverget. Compariso to q. si (/) As: Coverget. Limit compariso to l () 3 9. Determie the radius of covergece ad iterval of covergece of the give power series. ( ) x As: R =, I = (, ] b. x As: R =, I = (, )

10 c. ( ) x 3 As: R =, I = [, ] 0 x d. 3 As: R = [ 0, I = 0, ] 0 e. ( ) x As: R =, I = (, ) x f. 5 5 As: R = 5, I = [ 5, 5] ( ) x () ()! As: R =, I = (, ) ( ) (x 3) h. ( + ) As: R =, I = (, 4] i. 4(x + ) As: R = 4, I = ( 5, 3) (3x ) j. 3 [ As: R =, I = 3, 5 ) 3 k.!(x ) As: R = 0, I = l. (x 4) 3 + [ ]

11 As: R =, I = [3, 5] x m. (l ) = As: R =, I = [, ] 0. Fid a power series represetatio for the fuctio ad determie the radius ad iterval of covergece. f(x) = 3 x 4 As: f(x) = 3 ( + x 4 + x 8 + x + ) = 3 x 4 R =, I = (, ) b. f(x) = x + 0 As: f(x) = ( x ( ) 0 0 =0 ) R = 0, I = ( 0, 0) x c. f(x) = x + As: f(x) = ( ) x + R =, I = =0 (, ). Fid a power series represetatio to the give fuctio ad determie the radius of covergece. f(x) = l(5 x) As: f(x) = (l 5) x x 5, R = 5 b. f(x) = ( x) As: f(x) = ( + )x +, R = =0 c. f(x) = arcta(x/3) As: f(x) = ( ) (x/3)+ +, R = 3 =0 =0

12 . Fid the Maclauri series of f(x). Fid the radius of covergece of your result. f(x) = l( + x) ( ) + As: f(x) = x, R = b. f(x) = si(πx) As: f(x) = ( ) π + ( + )! x+, R = c. f(x) = xe x As: f(x) = =0 x ( )!, R = 3. Fid the Taylor series of f(x) cetered at the give value of a, ad fid the radius of covergece. f(x) = e x, a = 3 e 3 As: f(x) =! (x 3), R = =0 b. f(x) = si x, a = π/ As: f(x) = ( ) (x π/), R = ()! =0 c. f(x) = x, a = As: f(x) = ( ) ( + )(x ), R = =0 4. Estimate the give defiite itegral so that error < Aswer: cos ( x ) dx Area b. 0 Aswer: x si ( x 3) dx Area c. 0 e x dx

13 Aswer: Area * 5. Defie: { e /x if x 0 f(x) = 0 if x = 0 Prove that f is cotiuous at 0 b. Prove that f is differetiable at 0 c. Prove that f has derivatives of all order at 0 d. Fid the Taylor series of f cetered at 0 e. Show that the series you foud at (d) has a radius of covergece of f. Show that f is ot equal to its Taylor series at ay poit except the ceter. * 6 Defie: cos ( x ) f(x) = (Note that this fuctio f is defied by a ifiite series, =0 e but ot by a power series). Show that f is defied for all real umbers x by showig that the series is coverget for all real umbers x. b. Fid f (x) by term-by-term differetiatio (Eve though f is ot defied by a power series, but for this fuctio, you may still safely assume that f (x) may be obtaied from f by term-by-term differetiatio.). Show that the series you obtaied for f (x) this way is coverget for all real umbers x c. Usig the same method, fid the k-th derivative of f, f k (x) ad show that the series for f k (x) is coverget for all real umbers x, therefore f is ifiitely differetiable for all real umbers x. d. Fid the Maclauri series of f. e. Prove that your aswer i (d) has a radius of covergece of R = 0.

Math 106 Fall 2014 Exam 3.1 December 10, 2014

Math 106 Fall 2014 Exam 3.1 December 10, 2014 Math 06 Fall 0 Exam 3 December 0, 0 Determie if the series is coverget or diverget by makig a compariso DCT or LCT) with a suitable b Fill i the blaks with your aswer For Coverget or Diverget write Coverget