This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Review of Convergence Tests COMMENTS Divergence Test
t.) If klim uk = 0, Eu,= may or may no ‘1 .
‘+ﬂ converge, CincmdnSuiﬂ). lf lim uk 9% 0, then Eu; diverges. k~+= Let 214 be a series with positive terms, and let .
ﬂx} be the function that results when k is
replaced by .r in the formula for u... [ff is Use this test when f(x) is easy to Integral Test decreasing and continuous for x 2 I, then Integmte'
! ' *°’ This test onl a lies to series the
2 uk and ﬁx) dx 3’ P” I . . vi
I kg} 1 have posxtrve terms. '3 I both converge or both diverge. i Let 2a,: and 2!»; be series with nonnegative terms _
I such that Use this test as a last resort; other
I
9 Comparison Test tests are often easier to apply. () alsbla aZSbZoIsaksbks” . This test onl :1 lies to series
if 212,; converges. then 2m. converges. and 1f En;c . y. pp
. .  With nonnegatlve terms.
diverges, then Eb“ diverges. i
i Let Euk be a series with positive terms and 1 ! suppose . . uk+1 . .
Ratio Test f kin?! u = p Try this test when at Involves
m I " factorials or kth powers. {a} Series converges if p < l.
i 03) Series diverges ifp > I or p = +00. (c) No conclusion if p = I. 1
r Let Zuk be a series with positive terms such that i
i
i. p= lim (in
l k4+a Try this test when uk involves kth (a) Series converges if p < l. powers. (13) Series diverges if p > i or p = +06.
(c) No conclusion if p = l. Let Zak and Eb; be series with positive terms
such that This is easier to apply than the
comparison test, but still requires
some skill in choosing the series
Ebk for comparison. .. ' Limit Comparison Test
m If 0 < p < +00, then both series converge or both
diverge. p ﬂy“ a: + 03_— a; +5" 
; altern'atie.zr—_n‘as
It sass he were >. o forget * mag+a2a3+a4'4'.i_
convergeif' ' Q.
(a) d1>aa>a3_>" : ijm ak=tl 1M.  ' ' “i' 1:5: :"" itt: ' ' ‘ .i._ ,1 V : ‘ f3 ‘23:. _ .. ’9‘,“ _ ﬂ __ V I
'J—ﬁtﬁxﬂk! PB . sen, Wlti't nonzero W m _, _ _. ‘ ﬁ§5< fv‘ ' '7'? ' 'E“? "I;r " “Vi . _. a '  _ .
3‘3“? Tiesgm ? 7:le MEI The series need not have positive
Aggsolute' onvergence , ' ' ' 7‘. ,y  ' ' terms and need not be alternating (a) 'S‘eries‘conve'rges ebsomtely if p < l. . to use this test H ‘ (b)__ Sweeney's ifp >1 orp =; +én.
{c} ,No Conclqsipn‘ if P ='1}:.'. " VIRGINIA MATHEMATICAL SCHOLARS PROGRAM
Addendum to Review of ConvergenceTestS 1. Is the series geometric? Q
Does it have the form Zar’H with r < 1'? If so, then the sum of the series is exactly 1 a r .
n=l " If r S 1, then the series diverges. 2. Is the series telescoping?
Can the method of partial fractions be applied to at"?
If so, break it down and compute S", the nth partial sum.
Then recall that 2 an = 3,. by deﬁnition. 3. Is the given series a p—series? Does it look like 2 nip? The series converges as long as p > 1.
The series diverges if p S 1. ...
View
Full Document
 Spring '08
 varies
 Calculus

Click to edit the document details