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8.3. SERIES OF REAL NUMBERS
We then investigate the behavior of a given series by examining the sequence
S 1 , S 2 , . . . , S n , . . .
of its partial sums. If the sequence of partial sums converges to some finite number, then
we say that the corresponding series converges. Otherwise, we say the series diverges. From
our work in Activity 8.7, the series
∞
k=1
1
k 2
appears to converge to some number near 1.54977. We formalize the concept of convergence
and divergence of an infinite series in the following definition.
Definition 8.5. The infinite series
∞
k=1
a k
converges (or is convergent) if the sequence {S n } of partial sums converges, where
S n =
n
k=1
a k .
If lim n→∞ S n = S, then we call S the sum of the series
∞
k=1 a k . That is,
∞
k=1
a k = lim
n→∞
S n = S.
If the sequence of partial sums does not converge, then the series
∞
k=1
a k
diverges (or is divergent).
The early terms in a series do not contribute to whether or not the series converges or
diverges. Rather, the convergence or divergence of a series
∞
k=1
a k
is determined by what happens to the terms a k for very large values of k. To see why,
suppose that m is some constant larger than 1. Then
∞
k=1
a k = (a 1 + a 2 + · · · + a m ) +
∞
k=m+1
a k .