8.3. SERIES OF REAL NUMBERS
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(c) Use your result from (b) to decide whether the sequence
b n
n!
converges or
diverges. If the sequence
b n
n!
converges, to what does it converge? Explain
your reasoning.
2. There is a test for convergence similar to the Ratio Test called the Root Test. Suppose
we have a series
a k of positive terms so that a n → 0 as n → ∞.
(a) Assume
n
√
a n → r
as n goes to infinity. Explain why this tells us that a n ≈ r n for large values of n.
(b) Using the result of part (a), explain why
a k looks like a geometric series when
n is big. What is the ratio of the geometric series to which
a k is comparable?
(c) Use what we know about geometric series to determine that values of r so that
a k converges if
n
√ a n → r as n → ∞.
3. The associative and distributive laws of addition allow us to add finite sums in any
order we want. That is, if
n
k=0 a k and
n
k=0 b k are finite sums of real numbers, then
n
k=0
a k +
n
k=0
b k =
n
k=0
(a k + b k ).
However, we do need to be careful extending rules like this to infinite series.
(a) Let a n = 1 +
1
2 n and b n = −1 for each nonnegative integer n.
(i) Explain why the series
∞
k=0 a k and
∞
k=0 b k both diverge.
(ii) Explain why the series
∞
k=0 (a k + b k ) converges.
(iii) Explain why
∞
k=0
a k +
∞
k=0
b k
∞
k=0
(a k + b k ).
This shows that it is possible to have to two divergent series
∞
k=0 a k and
∞
k=0 b k but yet have the series
∞
k=0 (a k + b k ) converge.
(b) While part (a) shows that we cannot add series term by term in general, we
can under reasonable conditions. The problem in part (a) is that we tried to
add divergent series. In this exercise we will show that if
a k and
b k are
convergent series, then
(a k + b k ) is a convergent series and
(a k + b k ) =
a k +
b k .
(i) Let A n and B n be the nth partial sums of the series
∞
k=1 a k and
∞
k=1 b k ,
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(c) Use your result from (b) to decide whether the sequence
b n
n!
converges or
diverges. If the sequence
b n
n!
converges, to what does it converge? Explain
your reasoning.
2. There is a test for convergence similar to the Ratio Test called the Root Test. Suppose
we have a series
a k of positive terms so that a n → 0 as n → ∞.
(a) Assume
n
√
a n → r
as n goes to infinity. Explain why this tells us that a n ≈ r n for large values of n.
(b) Using the result of part (a), explain why
a k looks like a geometric series when
n is big. What is the ratio of the geometric series to which
a k is comparable?
(c) Use what we know about geometric series to determine that values of r so that
a k converges if
n
√ a n → r as n → ∞.
3. The associative and distributive laws of addition allow us to add finite sums in any
order we want. That is, if
n
k=0 a k and
n
k=0 b k are finite sums of real numbers, then
n
k=0
a k +
n
k=0
b k =
n
k=0
(a k + b k ).
However, we do need to be careful extending rules like this to infinite series.
(a) Let a n = 1 +
1
2 n and b n = −1 for each nonnegative integer n.
(i) Explain why the series
∞
k=0 a k and
∞
k=0 b k both diverge.
(ii) Explain why the series
∞
k=0 (a k + b k ) converges.
(iii) Explain why
∞
k=0
a k +
∞
k=0
b k
∞
k=0
(a k + b k ).
This shows that it is possible to have to two divergent series
∞
k=0 a k and
∞
k=0 b k but yet have the series
∞
k=0 (a k + b k ) converge.
(b) While part (a) shows that we cannot add series term by term in general, we
can under reasonable conditions. The problem in part (a) is that we tried to
add divergent series. In this exercise we will show that if
a k and
b k are
convergent series, then
(a k + b k ) is a convergent series and
(a k + b k ) =
a k +
b k .
(i) Let A n and B n be the nth partial sums of the series
∞
k=1 a k and
∞
k=1 b k ,