Barrons AP Calculus - David Bock

If there is a finite number S such that

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:

EXAMPLE 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 ∞.

B2. Theorems About Convergence or Divergence of Infinite Series.

The following theorems are important.