CHAPTER 37
Infinite Series
37.1 Prove that, if E an converges, then an=Q.
Let Then
37.2 Show that the harmonic series diverges.
etc. There-
fore, Alternatively, by the integral test,
37.3 Does imply that E an converges?
No. The harmonic series E 1/n (Problem 37.2) is a counterexample.
37.4 Let Sn = a + ar + •• • + ar" ', with r^l. Show that
rS=ar + ar2 + - - - + ar" + ar". S. = a + ar + ar2 + • • • + ar"~\ Hence, (r- 1)5_ = ar" - a = a(r" - I)
Thus,
37.5 Let a T^ 0. Show that the infinite geometric series and diverges if
By Problem 37.4, if M < i , 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.
37.6 Evaluate
By Problem 37.5, with
37.7 Evaluate
By Problem 37.5, with
37.8 Show that the infinite decimal 0.9999 • • • is equal to 1.
0.999 • • • by Problem 37.5, with
37.9 Evaluate the infinite repeating decimal d = 0.215626262
By Problem 37.5, with
Hence,
312
, INFINITE SERIES 313
37.10 Investigate the series
Hence, the partial sum
The series converges to 1. (The method used here is called "telescoping.")
37.11 Study the series
So
Thus, the series converges to
37.12 Find the sum of the series 4 — 1 + j — & + • • • .
This is a geometric series with ratio and first term a = 4. Hence, it converges to
37.13 Test the convergence of
This is a geometric series with ratio r = \ > 1. Hence, it is divergent.
37.14 Test the convergence of 3+I + I + I + - - - .
The series has the general term (starting with n = 0), but lim an = lim
Hence, by Problem 37.1, the series diverges.
37.15 Investigate the series
Rewrite the series as by Problems 37.11
and 37.10.
37.16 Test the convergence of
Hence, by Problem 37.1, the series diverges.
37.17 Study the series
So the partial sum
37.18 Study the series
Thus,
The partial sum
Infinite Series
37.1 Prove that, if E an converges, then an=Q.
Let Then
37.2 Show that the harmonic series diverges.
etc. There-
fore, Alternatively, by the integral test,
37.3 Does imply that E an converges?
No. The harmonic series E 1/n (Problem 37.2) is a counterexample.
37.4 Let Sn = a + ar + •• • + ar" ', with r^l. Show that
rS=ar + ar2 + - - - + ar" + ar". S. = a + ar + ar2 + • • • + ar"~\ Hence, (r- 1)5_ = ar" - a = a(r" - I)
Thus,
37.5 Let a T^ 0. Show that the infinite geometric series and diverges if
By Problem 37.4, if M < i , 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.
37.6 Evaluate
By Problem 37.5, with
37.7 Evaluate
By Problem 37.5, with
37.8 Show that the infinite decimal 0.9999 • • • is equal to 1.
0.999 • • • by Problem 37.5, with
37.9 Evaluate the infinite repeating decimal d = 0.215626262
By Problem 37.5, with
Hence,
312
, INFINITE SERIES 313
37.10 Investigate the series
Hence, the partial sum
The series converges to 1. (The method used here is called "telescoping.")
37.11 Study the series
So
Thus, the series converges to
37.12 Find the sum of the series 4 — 1 + j — & + • • • .
This is a geometric series with ratio and first term a = 4. Hence, it converges to
37.13 Test the convergence of
This is a geometric series with ratio r = \ > 1. Hence, it is divergent.
37.14 Test the convergence of 3+I + I + I + - - - .
The series has the general term (starting with n = 0), but lim an = lim
Hence, by Problem 37.1, the series diverges.
37.15 Investigate the series
Rewrite the series as by Problems 37.11
and 37.10.
37.16 Test the convergence of
Hence, by Problem 37.1, the series diverges.
37.17 Study the series
So the partial sum
37.18 Study the series
Thus,
The partial sum
