http://www.ck12.org Chapter 8. Infinite Series
sn=^109 ( 1 − 101 n)Now we find the limit of both sides:
n→lim+∞sn=nlim→∞^109(
1 − 101 n)
nlim→+∞^109(
1 − 101 n)
=nlim→∞^109 −n→lim+∞^109( 1
10 n)
=^109 − 0 =^109
The sum of the infinite series is^109 and so the series converges.
Geometric Series
Thegeometric seriesis a special kind of infinite series whose convergence or divergence is based on a certain
number associated with the series.
Geometric Series
A geometric series is an infinite series written as
a+ar+ar^2 +ar^3 +...+ar{i−^1 }+....In sigma notation, a geometric series is written as∑∞k= 1 ark−^1.
The numberris theratioof the series.
Example 4
Here are some examples of geometric series.
TABLE8.1:
Geometric Series a r1 +^14 + 412 + 413 +...+ 4 k^1 − 1 +... (^114)
−^56 + 652 − 653 +...+(−^56 )(−^16 )+
...
−^56 −^16
1 + 3 + 32 + 33 +...+ 3 k−^1 +... 1 3The convergence or divergence of a geometric series depends onr.
Theorem
Suppose that the geometric series∑∞k= 1 ark−^1 has ratior.
- The geometric series converges if|r|<1 and the sum of the series is 1 −ar.
- The geometric series diverges if|r|≥1.