Calculus Of One Real Variable By Pheng Kim Ving
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1. Infinite Series |
Consider the (infinite) sequence {an} = {a1, a2, a3, ...} of real numbers. The addition of all the terms of {an}:
We can have the index start from 1 by adjusting the
expression for the general term. In example 4, the kth
term corresponds
to index k 1. When the index starts from 0,
the term order, which starts from 1st, is greater than the index by 1. In
example
5, the kth term corresponds to index k + 1. When the index starts from 2, the term
order is less than the index by 1. When the
index starts from m > 1, the term order is
less than the index by m 1.
For a series, there are infinitely many terms to add. So if
we add the 1st term to the 2nd, then add the result to the 3rd, then
add the new result to the 4th, etc, by hand or even by a computer as we do for
an addition of finitely many terms, the addition
operation will go on forever. The solution is to resort to the concept of
limits. Let:
s1 = a1,
s2 = a1 + a2,
s3 = a1 + a2 + a3.
Each sk
is the sum of a part, namely the 1st k terms, of
the series, and so is called, well, a partial sum of the series. All the
sk's clearly form a
sequence, {sn} = {s1, s2, s3, ...}, which is called, well, the sequence of the
partial sums of the series. The
sum of the series, which is the result of the infinite addition, clearly is sn when n approaches infinity. Thus the sum of the
series is naturally defined as the limit of the sequence {sn}
of partial sums as n approaches
infinity.
For the rest of this section, when we say n we mean positive integer n.
Definition 1.1 (Infinite) Series
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Let {an}
be an infinite sequence of real numbers. The infinite series or the series
of real numbers associated with |
Remarks 1.1
2. The series is the addition of the terms of the
associated sequence, not the addition of the partial sums. It's the limit of
the
sequence of the partial sums.
3. A term of the series is a term of the associated sequence, not a partial sum.
4. The convergence of the series depends on the
convergence of the sequence of partial sums, not on the convergence of the
associated sequence.
5. For any n > 1, sn = sn1 + an.
Remark 1.2
A more mathematically precise definition of a series is that
a series is the sequence of its partial sums. In this way a series is
a real-valued function of the positive integers, because a sequence is so.
However in our work in this tutorial it's adequate to
think of a series as the addition or sum of all the terms of its associated
sequence.
Definition 1.2
Properties Of Series
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A series is positive if its associated sequence is
positive, ie if it has terms that are positive and may have terms that are 0.
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Example 1.1
Determine the convergence or divergence of each of the
following series. If it diverges, specify the type of divergence (diverges
to infinity or negative infinity or simply diverges).
{1.1} Example 2.1 Part c.
Solution
a. The nth partial sum is sn = 1 + 2 + ... + n = n(n + 1)/2, which approaches infinity as n approaches infinity. So the given
series diverges to infinity.
b. The nth partial
sum is sn = 1 + 1 + ... + 1 (n times) = n, which
approaches infinity as n approaches
infinity. So the given
series diverges to infinity.
c. The nth partial
sum is sn = 1/1,000,000
1/1,000,000 ... 1/1,000,000 (n times) = n/1,000,000, which approaches
negative infinity as n approaches infinity. So the given series
diverges to negative infinity.
d. The nth partial
sum is sn = 0 + 0 + ... + 0 (n times) = 0, which approaches 0 as n approaches infinity. So the given series
converges to 0.
e. The sequence of partial sums is {1, 0, 1, 0, 1, 0,
...}, which simply diverges. So the given series simply diverges.
EOS
For part e, it's easier to examine the sequence of
partial sums than to determine the expression for its general term, ie the
general nth partial sum.
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2. Some Useful Series |
The harmonic and geometric series, discussed below, are useful,
in the sense that their behavior (convergence or divergence)
is known and they're used in comparison tests for series, developed in the next
section. To determine the behavior of a series
we may compare it to a series whose behavior is known.
Harmonic Series
The series:
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Fig. 2.1 Harmonic Series Diverges To Infinity. |
Harmonic
Series
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The series: |
Telescoping
Series
Example 3.1
Calculate the sum of the following series if the series converges:
Solution
Decompose the general term into partial fractions:
EOS
The nth partial
sum of the series in the above example can be condensed or telescoped into an
expression involving fewer
terms of the partial sum. Thus such a series is called a telescoping series.
A Particular
Telescoping Series
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The solution in the above example uses the method of partial
fractions to try to calculate the sum of a telescoping series. When
the denominator of the general term is or can be factored as a product of 2 or
more factors involving n, we can try
to use the
method of partial fractions to decompose the general term into simpler
fractions.
Geometric Series
For the series:
Geometric
Series
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A series of the form: If r = 1 then sn = na. |
Note On Finite
Geometric Series
Finite Geometric
Series
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If r = 1 then the sum is na. |
Note that on the left-hand side of the above equation, there
are n terms and the highest exponent on r is n 1, while
on the
right-hand side, the power on r is n in the numerator and 1 in the denominator. The
highest exponent on r on the
left-hand
side is less than the power on r in the
numerator on the right-hand side by 1. For example we can also write:
If r = 1 then
the sum is (n + 1)a.
A Geometric
Series
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Determining A
Geometric Series
To determine the first term and the common ratio of a geometric series given in the form:
and make the same conclusion.
To determine if a series given as an addition of its first several terms followed by an ellipsis (...) is a geometric one:
divide the 2nd term by the 1st term, get say r, then check to see if every successive term
equals the previous term times r; if
this is true then the series is a geometric one, otherwise it's not.
Example 2.1
{2.1} Example 1.1 Part e.
Solution
a. Geometric series, first term a = 1, common
ratio r = 1/2, 1 < r
< 1,
EOS
Example 2.2
A ball bounces in such a way that the elapsed time to
complete a bounce, which is the time between any 2 consecutive strikes
on the floor, is always 2/5 of the time to complete the previous bounce.
Suppose the 1st bounce takes 3 seconds. Find the
length of time from when the ball first touches the floor till when it comes to
rest.
Solution
Let T be the
required length of time. Then:
The length of time from when the ball first touches the floor till when it comes to rest is 5 seconds.
EOS
Expressing A Repeating Decimal As A Quotient Of Integers
Example 2.3
Use the series method to express the repeating decimal 6.318181818... as a rational number (quotient of integers).
Solution
EOS
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3. Behaviors Of Series |
Necessary
Condition For Convergence
Theorem 3.1
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Proof
sn1 = a1 + a2 + ... + an1,
sn = a1 + a2 + ... + an1 + an,
sn sn1 = an.
EOP
Limit Of
Sequence And Divergence Of Series
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Example 3.1
Determine whether each series converges or diverges. Specify
the type of divergence (diverges to infinity or negative infinity or
simply diverges) if it diverges:
Solution
EOS
Convergence Depending
Only On Tail Of Series
For any series, the sum of its first N terms, where N is any (finite) positive integer, no matter how
large, is always a (finite)
number. So the convergence of a series depends only on its tail , which is always
infinitely long.
The following theorem states that a series converges iff its tail, no matter where it begins, converges.
Theorem 3.2
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This implies that the convergence or divergence of a series
isn't changed by a deletion or insertion or modification of any |
Proof
EOP
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4. Algebraic Operations On Series |
Given a series:
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Partial Sums Of
Algebra Of Series
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the nth partial sum of a difference of two series is the difference of the nth partial sums of the two original series. |
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Proof
EOP
If the sum of a series is infinity or negative infinity or doesn't
exist, multiplying the series by a non-0 constant will yield a sum
that still is infinity or negative infinity (depending on the signs of the
initial infinity and of the constant) or doesn't exist. This
observation is confirmed by the following corollary.
Corollary 4.1
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Proof
EOP
Combining Theorem 4.1 Part a and Corollary 4.1 we get:
Corollary 4.2
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Example 4.1
Calculate the sum of the following series it the series converges:
Solution
EOS
Example 4.2
Compute the sum of the following series it the series converges:
Solution
EOS
Example 4.3
Determine the convergence or divergence of the series:
Solution
EOS
Yes, the series:
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5. Series And Using Calculator Or Computer |
Problems & Solutions |
1. Consider the series: 1 + 0 1 + 0 + 1 + 0 1 + 0 + 1 + 0 1 + 0 +
... Find the following partial sums: s1, s2, s3, s4, s5, s14,
s63, and s800.
Solution
The sequence of partial sums is: {1, 1, 0, 0, 1, 1 , 0, 0, 1, 1, 0, 0, ...}. So:
Thus s1 = 1, s2 = 1, s3 = 0, s4 = 0, s5 = 1, s14 = 1, s63 = 0, and s800 = 0.
2. Find the first 5 terms a1, a2, ..., a5 of the series that has as sequence of partial sums:
Solution
3. For each of the following series, find its sum if its converges, or
show that it diverges and specify the type of its divergence
(diverges to infinity or negative infinity or simply diverges).
Solution
a. Geometric series, first term a = 1, common ratio r = 1/3, so:
4. Determine the behavior of this telescoping series (convergence or
divergence). If it converges, find its sum. If it diverges,
specify the type of divergence
(diverges to infinity or minus infinity or simply diverges).
Solution
Decompose into partial fractions:
5. Does the following series converge or diverge? If
it converges, calculate its sum. If it diverges, determine the type of
divergence (diverges to infinity or
negative infinity or simply diverges).
Solution
6. Use the series method to find the rational representation of the repeating decimal 42.21847847847...
Solution
7. A ball is dropped from a height of 3 m. Each time it
hits the ground, it bounces back up to a height 2/3 of that from which it
fell. Assuming the ball is allowed
to bounce up and down indefinitely, find the total distance it travels before
coming to rest.
Solution
Let d be the
required distance. Then:
The total distance the ball travels before coming to rest is 15 m.
8. Is the following statement true or false? If it's true, prove it. If it's false, give a counter-example showing its falsehood.
Solution
False. Counter-example:
9. A humorous math professor offers to teach you an introductory-calculus
course one-on-one for 15 sessions for 3 hours per
session for just $1 for the first
session, provided you agree to hire him for the total of 15 sessions, 1 session
per week
during 15 consecutive weeks, and to
double his fee each session. Before you accept his offer , you should do the following.
a. Find the total amount of money you would pay him for the 15 sessions.
b. Determine the average fee per hour you would pay him.
Solution
a. Let A be the total amount of money that would be paid to the professor. We have:
The average fee per hour that would be paid to him is approximately $728.16/hr.
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