Calculus Of One Real Variable – Chapter 9 – Section 9.2

9.2
Infinite Series

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1. Infinite Series

Consider the (infinite) sequence {a_n} = {a₁, a₂, a₃, ...} of real numbers. The addition of all the terms of {a_n}:

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 s_k 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
s_k's clearly form a sequence, {s_n} = {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 s_n when n approaches infinity. Thus the sum of the
series is naturally defined as the limit of the sequence {s_n} 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

Let {a_n} be an infinite sequence of real numbers. The infinite series or the series of real numbers associated with
sequence {a_n} is the infinite addition of all the terms of {a_n}, and thus is:

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, s_n = s_n–1 + a_n.

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

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.
Similarly for the properties strictly positive, negative, and strictly negative, and for the adverb ultimately.

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} Example 2.1 Part c.

Solution
a. The nth partial sum is s_n = 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 s_n = 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 s_n = – 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 s_n = 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:

Fig. 2.1 Harmonic Series Diverges To Infinity.

Harmonic Series

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

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

A series of the form:

If r = 1 then s_n = na.

Note On Finite Geometric Series

Finite Geometric Series

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

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

^{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

Proof
s_n–1 = a₁ + a₂ + ... + a_n–1,
s_n = a₁ + a₂ + ... + a_n–1 + a_n,

s_n – s_n–1 = a_n.

EOP

Limit Of Sequence And Divergence Of Series

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

This implies that the convergence or divergence of a series isn't changed by a deletion or insertion or modification of any
finite number of terms.

Proof

EOP

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4. Algebraic Operations On Series

@@ change order to: sum, diff, const mult; def sum intuitively first

Given a series:

Definition 4.1

Partial Sums Of Algebra Of Series

  the nth partial sum of a difference of two series is the difference of the nth partial sums of the two original series.

Theorem 4.1

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

Proof

EOP

Combining Theorem 4.1 Part a and Corollary 4.1 we get:

Corollary 4.2

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:

@@ write ans in form integer followed by fraction

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

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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.

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2. Find the first 5 terms a₁, a₂, ..., a₅ of the series that has as sequence of partial sums:

Solution

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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:

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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:

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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

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6. Use the series method to find the rational representation of the repeating decimal 42.21847847847...

Solution

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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.

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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:

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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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