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Unformatted text preview: Workshop 8
1) Each of the following sequences has limit 0:
n ∞ 1
n n=1 ∞ ∞ 1
n2 n=1 n=1 1
n=1 a) For each sequence, state exactly how large n must be to ensure that the
term an of the sequence (and all later terms as n increases) satisfy |an | < 10−4 .
b) Similarly, how large must n be to ensure that |an | < 10−8 ?
c) Use this information to explain which sequence approaches 0 most rapidly
and which approaches 0 least rapidl
2) a) Two students are sharing a loaf of bread. Student Alpha eats half of
the loaf, then student Beta eats half of what remains, then Alpha eats half
of what remains, and so on. How much of the loaf will each student eat?
b) Two students are sharing a loaf of bread. Student Alpha, now hungrier
and more ferocious, eats two-thirds of the loaf, then student Beta eats eats
half of what remains, then Alpha eats two-thirds of what remains, then Beta
eats half of what remains, and so on. How much of the loaf will each student
c) Now start with three students: Alpha, Beta, and Gamma. They decide
to share a loaf of bread. Alpha eats half of the loaf, passes what remains to
Beta who eats half, and then on to Gamma who eats half, and then back to
Alpha who eats half, and so on. How much of the loaf will each student eat?
3) Consider the following sequences:
an = 1 +
n n ; 1
bn = 1 + 2
n n ; 1
cn = 1 + √
n n . a) Use your calculator to plot the ﬁrst ten terms of each of these sequences.
Then use this information to guess the limiting behavior of each of the sequences.
b) Replace n by x and use L’Hopital’s Rule to ﬁnd the limit of each as x
tends to inﬁnity.
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This note was uploaded on 02/29/2012 for the course MATH 152 taught by Professor Sc during the Fall '08 term at Rutgers.
- Fall '08