314
CHAPTER 37
37.19
37.20
37.21
37.22
37.23
37.24
37.25
37.26
Study the series
Hence, the partial sum
Study the series
So
Evaluate
So the partial sum
is either
or
In either case, the partial sum approaches 1.
Evaluate
The partial sum
Evaluate
so, by Problem 37.1, the series diverges.
Evaluate
The series diverges.
Find the sum
and show that it is correct by exhibiting a formula which, for each e > 0, specifies
an integer m for which \S n — S\< e holds for all n > m (where S n is the nth partial sum)
is a geometric series with ratio r = 5 and first term a = 1. So the sum
In fact,
assume e > 0. Then, by Problem 37.4, S n = 1
Now
We want
Choose m to be the least positive integer that exceeds
Determine the value of the infinite decimal 0.666 + • • -.
is a geometric series with ratio
and first term
Hence.
the sum is
The partial sum