then we say that infinite series is convergent, or converges to S, or has the sum S,
and we write, in this case,
When there is no source of confusion, the infinite series (1) may be indicated
simply by
Example 10 __
Show that the geometric series . . . converges to 2.
SOLUTION: Let S represent the sum of the series; then:
Subtraction yields
Hence, S = 2.
BC ONLYExample 11 __
Show that the harmonic series . . . diverges.
SOLUTION: The terms in the series can be grouped as follows:
This sum clearly exceeds
which equals
Since that sum is not bounded, it follows that diverges to ∞.