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8.3. SERIES OF REAL NUMBERS
(b)
k 3 + 2
k 2 + 1
(c)
10 k
k!
(d)
k 3 − 2k 2 + 1
k 6 + 4
⊳
Summary
In this section, we encountered the following important ideas:
• An infinite series is a sum of the elements in an infinite sequence. In other words, an
infinite series is a sum of the form
a 1 + a 2 + · · · + a n + · · · =
∞
k=1
a k
where a k is a real number for each positive integer k.
• The nth partial sum S n of the series
∞
k=1 a k is the sum of the first n terms of the series.
That is,
S n = a 1 + a 2 + · · · + a n =
n
k=1
a k .
• The sequence {S n } of partial sums of a series
∞
k=1 a k tells us about the convergence
or divergence of the series. In particular
- The series
∞
k=1 a k converges if the sequence {S n } of partial sums converges. In
this case we say that the series is the limit of the sequence of partial sums and
write
∞
k=1
a k = lim
n→∞
S n .
- The series
∞
k=1 a k diverges if the sequence {S n } of partial sums diverges.
Exercises
1. In this exercise we investigate the sequence
b n
n!
for any constant b.
(a) Use the Ratio Test to determine if the series
10 k
k! converges or diverges.
(b) Now apply the Ratio Test to determine if the series
b k
k! converges for any
constant b.