13 - 5 Sums of Infinite
Series
*In this section,
we discuss
the sum of infinite Geometric Series only. A series can converge or diverge. A series
that converges has a finite
limit,
that is a number that is approached. A
series that diverges means either the partial sums have no limit or
approach
infinity. The difference is in the
size
of the common ratio. If |r| < 1,
then the series will converge. If |r| *__>__
1
then the series diverges. (Piece of
cake! Time to converge
on
a few series! Very punny!!)
__SUM
OF AN INFINITE GEOMETRIC SERIES__
*If |r| < 1,
the infinite
Geometric Series:*
*t*_{1}
+ t_{1}r + t_{1}r^{2}
+ ^{. . . }+ t_{1}r^{n}
+ ^{. . .}
*converges to the
sum*^{ }^{ }s^{ }=
t_{1 }/
(1 -
r).
*If |r| *__>__
1, and
t_{1} does not =
0, then
the series diverges.
*It's time
for the
diverge/converge
game!! Drum roll please! Tell whether each series converges
or diverges. If it converges, find the sum!*
*Sample problems*
*1) 1 + 2 +
3 +
4 + . . .*
*2) 1/2 +
1/4 +
1/8 + 1/16 + . . .*
*3) 1 + 1/3
+ 1/9
+ 1/27 + . . .*
*4) 1 + 0.1 + .01 + .001 + .0001 +
. .
.*
*(answers
are down
the
page with the problems worked out! Follw the thumbs)*
**Here's the answers!!**
**1) r = 1, so the
series
diverges. (That was easy!)**
**2) r = 1/2, so
this series
converges. Apply the sum and voila!**
**s =
.5/(1 - .5) = .5/.5 = 1 (No
challenge)**
**3) r = 1/3,
so this
one also converges. As above, use the sum and:**
**s = 1/(1 - 1/3) = 1/(2/3) = 3/2 = 1.5
(Easy)**
**4) r = .1,
so again
this one converges. Do the same as 2 and 3**
**S =
1/(1 - .1) = 1/.9 = 10/9 = 1 and 1/9 (Yikes)**

__Other problems__
*1) For what
values
of x does the following infinite series converge?*
*1 + (x - 3) + (x
- 3)*^{2}
+ (x - 3)^{3} + ^{. . .
}
** Dividing
the second term by the first term gives the ratio to be (x - 3).
Using the fact that it must converge only if |r| < 1 and our r = (x
- 3), leads the road runner to conclude that |x - 3| < 1.
Remembering
your absolute value facts**
**-1 < x - 3 <
1**
**2 < x < 4**
** This
interval is called the interval
of convergence. It means that any
value
in this interval causes the series to converge and any value ( say 5)
will
cause the interval to diverge. **

**2) The
repeating decimal
. 27272727 . . . can be written as an infinite series. Write it
as
a series and tell if it diverges/converges. If it converges, find
the sum.**

** Write it as .27 + .0027 + .000027
. . .**

**It's now an infinite series.
Mickey spots
the common ratio of the series as .0027/.27 = .01. Therefore, it
converges!! He use the formula and presto:**
**S = .27/(1 - .01)
= .27/.99
= 27/99 = 3/11**

We are all set
to tackle a new subject and a new mathematical
symbol.
Yes, that's
right, we are talking about Sigma
Notation!