CHAPTER 37
Infinite Series
37.1
Prove that, if E a n converges, then
a n =Q.
Let
Then
37.2
Show that the harmonic series
diverges.
fore,
etc.
ThereAlternatively, by the integral test,
37.3
Does
imply that E a n converges?
No. The harmonic series E 1/n (Problem 37.2) is a counterexample.
37.4
37.5
37.6
37.7
37.8
37.9
312
Let S n = a + ar + •• • + ar" ', with r^l. Show that
rS=ar + ar
2 + --- + ar" + ar". S. = a + ar + ar
2 + • • • + ar"~\ Hence, (r- 1)5_ = ar" - a = a(r" - I)
Thus,
Let a T^ 0. Show that the infinite geometric series
and diverges if
By Problem 37.4,
if M since r"-*0; if \r\>\,
JSJ—»+°°, since |r| -*+<». If r = l, the series is a + a + a H
, which diverges since a¥=0. If
r = — 1, the series is a — a + a — a + • • • , which oscillates between a and 0.
Evaluate
By Problem 37.5, with
Evaluate
By Problem 37.5, with
Show that the infinite decimal 0.9999 • • • is equal to 1.
0.999 • • •
by Problem 37.5, with
Evaluate the infinite repeating decimal d = 0.215626262
By Problem 37.5, with
Hence,