Page 261 - Calculus Workbook For Dummies

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Chapter 13: Infinite Series: Welcome to the Outer Limits

Testing Three Basic Series

In this section, you figure out whether geometric series, p-series, and telescoping
series are convergent or divergent.
3
n
Geometric series: If < r0 < 1, the geometric series ! ar converges to a .
n 0 1 - r
=
If r $ 1, the series diverges. Have you heard the riddle about walking halfway to
the wall, then halfway again, then half the remaining distance, and so on? Those
steps make up a geometric series.
p-series: The p-series ! 1 p converges if >p 1 and diverges if p # 1.
n
Telescoping series: The telescoping series, written as h 1 - h 2 + _ h 2 - h 3 +
_
i
i
i
_ h 3 - h 4 + ... + _ h n - h n 1+ i, converges if h n 1+ converges. In that case, the series
converges to h 1 - limh n 1+ . If h n 1+ diverges, so does the series. This series is very
n " 3
rare, so I won’t make you practice any problems.
When analyzing the series in this section and the rest of the chapter, remember that
multiplying a series by a constant never affects whether it converges or diverges. For
3
3
example, if ! u n converges, then so will 1000$ ! . Disregarding any number of initial
u n
3
n 1
3
=
=
n 1
terms also has no affect on convergence or divergence: If ! u n diverges, so will ! .
u n
n 1 n 982
=
=
Q. Does 1 + 1 + 1 + 1 + 1 + ... converge or Q. Does ! 1 converge or diverge?
2 4 8 16 n
diverge? And if it converges, what does it A. ! 1 is the p-series ! 1 1
converge to? n / 1 2 where p = 2 .
n
A. Each term is the preceding one multiplied Because <p 1, the series diverges.
1
by . This is, therefore, a geometric
2
series with r = 1 . The first term, a,
2
equals one, so the series converges to
a = 1 = 2.
1 - r 1
1 -
2