---

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: t₁ + t₁r + t₁r² + ^{. . .} + t₁rⁿ + ^{. . .} converges to the sum  s = t₁ / (1 - r). If |r| > 1, and t₁ 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 + ^{. . .}
  Beep!  Beep!  The road runner has the answer!  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 his absolute value facts (he had Mr. K too!), beep beep recalls -1 < x - 3 < 1 2 < x < 4 The road runner is one heck of a mathematician!  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.  ( I'm sure you guys are as clever as the road runner!  I remember him from class!  You are much smarter!
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.
Mickey, also a former student, knows how to do this one.  Mickey knows enough math to 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 (remarkable achievement considering he's a mouse!) And now it's time to say goodbye to all our company!  No, wait, that's a different show!  (Showing my age here!)  We are all set to tackle a new subject and a new mathematical symbol.    Yes, that's right, we are talking about Sigma Notation!  Same time, same channel!  (This must be an age thing!)