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**Unformatted text preview: **Math 21C
Kouba
Discussion Sheet 10 1.) Determine if each of the following series, which contain both positive and negative
terms, is conditionally convergent, absolutely convergent, or divergent. Remember what
the absolute convergence test says. °° 1 °° 1 °° n + 1
. E —1 “+1 b. —1 "+1 . E —1 “+1 U
1
derivative to show that f = 1;: :— 9 decreases after some point.) °° 2 50 5 5 5 5 5 5 5 5
d.) Z(—1)n+1:5:50 e. n 1 0° 3 1 3
f.) Z(—1)" (1n n) (Use a derivative to show that f = ( If) decreases after
n n 1
some point.) 2.) Use the absolute ratio test to determine the interval of convergence for each series. 00 $77. 00 $77. 00 (l. _ 1)n+1 0° :LJL—l
. . . —1 n— d. — a)7;1n+1 b):4:41n2+1 C);( ) n2" );(n+1)!
00 2 n 00 (n!)m" 0° n+1 nn e.) E as f.) E M g.) E n + 3 :1: (HINT: Use the absolute n
71:]. 71:0 n=1
root test.) The following three problems illuminate the odd behavior of some inﬁnite series, which
contain both positive and negative terms. At the very least we are made aware of the need
to use precise testing methods (e.g., alternating series test, absolute convergence test,
sequence of partial sums test, geometric series test, etc.), and that our intuition sometimes
may fail when applied to problems like these. 00 3.) a.) Consider the series Z(cos(2mr) + cos((2n + 1)7r)) .
71.21
i.) Write out the ﬁrst 4 terms of the series. ii.) Use a particular test to determine if the series converges or diverges. 00 b.) Consider the series 23—1)"+1 . n21 1.) Write out the ﬁrst 8 terms of the series.
ii.) Use a particular test to determine if the series converges or diverges. c.) The series in a.) and b.) appear to be the same, but the answers in a.) ii.) and
b.) ii.) show otherwise. 00
4. . C ‘01 th ‘ —1"+1 n -
) a.) onsr er e series ) n+ 1 i.) Write out the ﬁrst 8 terms of the series.
ii.) Use a particular test to determine if the series converges or diverges. b) C 'd the ' i 2" _ 1 2”
. 01181 61' series "— .
“:1 2n 2n + 1 i.) Write out the ﬁrst 4 terms of the series.
ii.) Use a particular test to determine if the series converges or diverges. c.) The series in a.) and b.) appear to be the same, but the answers in a.) ii.) and
1).) ii.) show otherwise. 5.) a.) Consider the series Z(~1/2)" . 11:0
i.) Write out the ﬁrst 8 terms of the series. ii.) Use a particular test to determine if the series converges or diverges. °° 1 1
b.) Consider the series E — 22"“) .
n~0 i.) Write out the ﬁrst 4 terms of the series.
ii.) Use a particular test to determine if the series converges or diverges. c.) Evaluate each series and then their sum : 1 1 1 1 1 1 1 1
(+4+16+EZ+W>+( _ _ T _ T _ _ — _ ' d.) The series in a.), b.), and c.) appear to be the same, and the answers in a.) ii.),
b.) ii.), and c.) bear out that they are. What characteristic do the series in 5.) have
that those in 3.) and 4.) do not ? The series (All are the same.) in 5.) is an absolutely
convergent series, which converges to the same number no matter how the terms in the series are grouped or in what order they are added. This is not always true of conditionally
convergent or divergent series. “Nothing in life is to be feared. It is only to be understood.” 7 Marie Curie ...

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