Infinite Series solved questions

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