Let be given a sequence
of real numbers. The expression
is called the infinite series
as its general term
whose general term is
is called the sequence of partial sums
of the given series.
For the definition of the convergence of a sequence, v.s. Def. 6.1.
if its sequence of partial sums is
convergent. Otherwise it is divergent
if the sequence of partial sums converges, its limit is called the sum of the series.
. The number x is the sum of a geometric series, whose first term is equal to
and whose ratio is equal to
(A Telescoping Series)
Consider the series whose general term is
. we have:
It follows that the series
. is convergent and its sum
is equal to 1.
We consider the series
. If it is convergent, then
Once again, consider the telescopic series from Example 1.6
. We have:
Take the series:
Despite the fact that the general term of the series has a limit equal to 0, this series is divergent
(it has an infinite sum).
(The harmonic series)
Take the series
We prove that this series is divergent by grouping the terms in the following way: