13 - 5 Sums of Infinite
In this section,
the sum of infinite Geometric Series only. A series can converge or diverge. A series
that converges has a finite
that is a number that is approached. A
series that diverges means either the partial sums have no limit or
infinity. The difference is in the
of the common ratio. If |r| < 1,
then the series will converge. If |r| >
then the series diverges. (Piece of
cake! Time to converge
a few series! Very punny!!)
OF AN INFINITE GEOMETRIC SERIES
If |r| < 1,
+ t1r + t1r2
+ . . . + t1rn
+ . . .
converges to the
sum s =
If |r| >
t1 does not =
the series diverges.
game!! Drum roll please! Tell whether each series converges
or diverges. If it converges, find the sum!
1) 1 + 2 +
4 + . . .
2) 1/2 +
1/8 + 1/16 + . . .
3) 1 + 1/3
+ 1/27 + . . .
4) 1 + 0.1 + .01 + .001 + .0001 +
page with the problems worked out! Follw the thumbs)
Here's the answers!!
1) r = 1, so the
diverges. (That was easy!)
2) r = 1/2, so
converges. Apply the sum and voila!
.5/(1 - .5) = .5/.5 = 1 (No
3) r = 1/3,
one also converges. As above, use the sum and:
s = 1/(1 - 1/3) = 1/(2/3) = 3/2 = 1.5
4) r = .1,
this one converges. Do the same as 2 and 3
1/(1 - .1) = 1/.9 = 10/9 = 1 and 1/9 (Yikes)
1) For what
of x does the following infinite series converge?
1 + (x - 3) + (x
+ (x - 3)3 + . . .
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.
your absolute value facts
-1 < x - 3 <
2 < x < 4
interval is called the interval
of convergence. It means that any
in this interval causes the series to converge and any value ( say 5)
cause the interval to diverge.
. 27272727 . . . can be written as an infinite series. Write it
a series and tell if it diverges/converges. If it converges, find
Write it as .27 + .0027 + .000027
. . .
It's now an infinite series.
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 = 3/11
We are all set
to tackle a new subject and a new mathematical
right, we are talking about Sigma