Chapter 9 – Infinite Series

Sequence
• Definition of a sequence –
An infinite sequence, or a sequence, is an unending succession of numbers, called terms. Usually we
denote the first term as 𝑎1, the second as 𝑎2 , and so on. For example

Each of these sequences has a definite pattern, so we have a rule

or formula for generating the terms.
o Find the general term of the sequence

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It is also common to write only the general term enclosed in braces like
Thus,

The letter 𝑛 is called the index for the sequence. Moreover, it is not essential to start the index at 1;
sometimes it is more convenient to start it at 0. For example

can be written as or

• Limit of Sequence –
Since sequences are functions, we can inquire about their limits. Because a sequence {𝑎𝑛 } is only
defined for integer values of 𝑛, the only limit that makes sense is the limit of 𝑎𝑛 as 𝑛 → +∞.

We say that a sequence {𝑎𝑛 } approaches a limit 𝐿 if the terms in the sequence eventually become
arbitrary close to 𝐿.
 Geometrically, it means that for any positive number 𝜀 there is a point in the sequence after which
all terms lie between the lines 𝑦 = 𝐿 − 𝜀 and 𝑦 = 𝐿 + 𝜀.

Here are some properties of limits that can apply to sequences.

If the general term of a sequence is 𝑓(𝑛), where 𝑓(𝑥) is a function defined on the entire interval
[1, +∞), then the values of 𝑓(𝑛) can be viewed as “sample values” of 𝑓(𝑥) taken at the positive integers.

But the converse is not true. We cannot infer that 𝑓(𝑥) → 𝐿 as 𝑥 → +∞ from the fact that 𝑓(𝑛) →
𝐿 as 𝑛 → +∞.

o In each part, determine whether the sequence converges or diverges by examining the limit
as 𝑛 → +∞.

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o In each part, determine whether the sequence converges, and if so, find its limit.

Converge to0
o Find the limit of the sequence

O
 𝑛
o Show that lim √𝑛 = 1
𝑛→+∞

o Write out the first five terms of the sequence, determine whether the sequence converges,
and of so find its limit.

a) d)

b) e)

c) f)

o Find the general term of the sequence, starting with 𝑛 = 1, determine whether the sequence
converges, and if so find its limit.

a) c)

b) d)

The sequence converges to 0, since the even-numbered terms and the

odd-numbered terms both converge to 0.

The sequence diverges, since the odd-numbered terms converge to 1 and

the even-numbered terms converge to 0.
• The Squeezing Theorem for Sequences –

o Consider the sequence , does it converge or diverge?

Convenge

Infinite Series
• Sums of Infinite Series

Consider 0.3333333333,
We view this as 0.3 + 0.33 + 0.333 + 0.3333 + ….
3 3 3 3
Or 10
+ 102 + 103 + 104 + ⋯

The sequence of numbers 𝑠1, 𝑠2, 𝑠3, 𝑠4, … can be viewed as a succession of approximations to the “sum”
of the infinite series, which we want to be 1/3. To see that it actually foes to 1/3, we must calculate the
limit of the general term in the sequence of approximations,
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Since → 0 as 𝑛 → +∞, it follows that
10𝑛

Which we denote by writing

- So the general concept of the “sum” of an infinite series

We denote 𝑠𝑛 as the sum of the initial terms of the series, up to 𝑛.

- The number 𝑠𝑛 is called the nth partial sum of the series.

- The sequence {𝑠𝑛 }+∞
𝑛=1 is called the sequence of partial sums.

o Determine whether the series 1 − 1 + 1 − 1 + 1 − 1 + ⋯ converges or diverges. If

converges, find the sum.
 • Geometric Series :
If the initial term of the series is 𝑎 and each term is obtained by multiplying the preceding term by 𝑟,
then the series has the form

Such series are called geometric series, and the number 𝑟 is called the ratio for the series.
For example:

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o In each part, determine whether the series converges, and if so fins its sum.

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o Find the rational number represented by the repeating decimal 0.784784784 ….

o In each part, find all values of 𝑥 for which the series converges, and find the sum of the
series for those values of 𝑥.
o Determine whether the series converges, and if so find its sum.

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 Convergence Tests
• The Divergence Test :
Note that the 𝑘𝑡ℎ term in an infinite series ∑ 𝑢𝑘 is called the general term.

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An alternative form for part (a) is :

o Determine whether the series converges or diverges.

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• Algebraic Properties of Infinite Series :

o Find the sum of the series

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o Determine whether the following series converge or diverge

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 • The Integral Test :
The expressions

are related in that the integrand in the improper integral results when the index 𝑘 in the general term
of the series is replaced by 𝑥 and the limits of summation in the series are replaced by the
corresponding limits of integration.

o Show that the integral test applies, and use the integral test
to determine whether the following series converge or diverge.

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One of the most important of all diverging series is the harmonic series.

Pf:

• 𝒑 – series :
A 𝑝 – series is an infinite series of the form , , where 𝑝 > 0.

For example: (The case 𝑝 = 1 is the harmonic series)

 o Show that diverges.

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o Determine whether the series converges.

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• The Comparison Test :

The underlying idea of this test is to use the known convergence or divergence of a series to deduce
that of another series.

Pf: Think of it by interpreting the terms in the series as areas of rectangles.

- Using the Comparison Test

 o Use the Comparison Test to determine whether the following series converge or diverge.

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Comparison Test :
It is not always so straightforward to find the series required for comparison, so we will consider an
alternative to the comparison test.

(The case 𝑝 = 0 or 𝑝 = +∞ will later be discussed.)

o Determine whether the following series converge or diverge.

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The Ratio Test :
The Comparison Test and the Limit Comparison Test hinge on first making a guess about
convergence and then finding an appropriate series for comparison, both of which can be difficult
tasks.

o Use Ratio Test to determine whether the following series converge or diverge.
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 • The Root Test :
In cases where it is difficult or inconvenient to find the limit required for the Ratio Test.

o Use the Root test to determine whether the following series converge or diverge.

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• Alternating Series :
Series whose terms alternate between positive and negative are called alternating series.

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o Use the Alternating Series Test to show that the following series converges.

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 • Approximating Sums and the error bound :

o Given that

a) Find an upper bound on the magnitude of the error that results if 𝑙𝑛2 is approximated
by the sum of the first eight terms in the series.
b) Find a partial sum that approximates 𝑙𝑛2 to one decimal-place accuracy.

• Absolute Convergence :
Consider the series It does not fit into any categories
studied so far.
So we have:

o Determine whether the following series converge absolutely.

Note that in part (b) above, it converges, since it is the alternating harmonic series, yet, it does not
converge absolutely. But, if a series converges absolutely, then it converges.
 o Show that the series converge.

• Conditional Convergence :
Now, we will discuss about the convergence or divergence of a series that diverges absolutely. For
example :

Both of these series diverge absolutely, since in each case the series of absolute values is the
divergent harmonic series.
However, the first case actually converges, and the second case diverges.
Thus, we name:
A series that converges but diverges absolutely is to converge conditionally.

o Determine whether this series converges absolutely or converges conditionally.

• The Ratio Test for Absolute Convergence :

o Determine whether the series converges.

• Maclaurin and Taylor Polynomials :
Here, we will consider how we can improve on the accuracy of local linear approximations by using
higher-order polynomials as approximating functions.

Local Quadratic Approximation ( Will not be on the Exam)

Recall that local linear approximation 𝑓(𝑥 ) ≈ 𝑓 (𝑥0 ) + 𝑓′(𝑥0)(𝑥 − 𝑥0 ), now if we bend the
approximation in the direction of the exact function, we expect it to be a curve close to the graph of 𝑓.
So we have the approximation form:
𝑓 (𝑥 ) ≈ 𝑐0 + 𝑐1 𝑥 + 𝑐2 𝑥 2, and the values of 𝑐0 , 𝑐1 , 𝑐2 must be chosen so that the approximation function
must be 𝑝(𝑥 ) = 𝑐0 + 𝑐1 𝑥 + 𝑐2 𝑥 2 and its first two derivatives match those of 𝑓 at 0. Thus, we want
𝑝(0) = 𝑓 (0), 𝑝′ (0) = 𝑓 ′ (0), 𝑝′′ (0) = 𝑓′′(0)
The best values of 𝑝(0), 𝑝′ (0), 𝑎𝑛𝑑 𝑝′′(0) are as follows:

Thus, it follows that

So we have

• Maclaurin Polynomials :
Given that a function 𝑓 can be differentiated 𝑛 times at 𝑥 = 𝑥0, find a polynomial 𝑝 of degree 𝑛
with the property that the value of 𝑝 and the values of its first 𝑛 derivatives match those of 𝑓 at 𝑥0
 o Find the Maclaurin Polynomials 𝑝0 , 𝑝1 , 𝑝2 , 𝑝3 , 𝑎𝑛𝑑 𝑝𝑛 for 𝑒 𝑥 .

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 • Taylor Polynomials :
Now, we have focused on approximating a function 𝑓 in the vicinity of 𝑥 = 0. Now, we will
consider a more general case of approximating. We still want an 𝑛th degree polynomial 𝑝 with the
property that its value and the values of its first 𝑛 derivatives match those of 𝑓 at 𝑥0. But we
simplify the computations if we express it in powers of 𝑥 − 𝑥0, that is

And 𝑥0 = 0 shows that

Thus, we have:

o Find the first four Taylor polynomials for 𝑙𝑛𝑥 about 𝑥 = 2.

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 • Maclaurin and Taylor Series :
We just extend the notions of Maclaurin and Taylor polynomials to series by not stopping the
summation index of 𝑛. Thus we have the following:

o Find the Maclaurin series for

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o Find the Taylor series for 1/𝑥 about 𝑥 = 1.

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• Power Series in 𝒙 :
We define, if 𝑐0 , 𝑐1 , 𝑐2 , … are constants and 𝑥 is a variable, then a series of the form

is called a power series in 𝒙.

Examples:

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True, every Maclaurin series

is a power series in 𝑥.

• Radius and Interval of Convergence :

If a numerical value is substituted for 𝑥 in a power series ∑ 𝑐𝑘 𝑥 𝑘 , then the resulting series of
numbers may either converge or diverge. We need to determine the x-values for which a given
power series converges; that is called its convergence set.

The convergence set of a power series in 𝑥 is called the interval of convergence.

In the case where the convergence set is :
𝑥 = 0 ; 𝑤𝑒 𝑠𝑎𝑦 𝑡ℎ𝑒 𝑠𝑒𝑟𝑖𝑒𝑠 ℎ𝑎𝑠 𝒓𝒂𝒅𝒊𝒖𝒔 𝒐𝒇 𝒄𝒐𝒏𝒗𝒆𝒓𝒈𝒆𝒏𝒄𝒆 𝟎
{ ( −∞, ∞) ; 𝑤𝑒 𝑠𝑎𝑦 𝑡ℎ𝑒 𝑠𝑒𝑟𝑖𝑒𝑠 ℎ𝑎𝑠 𝒓𝒂𝒅𝒊𝒖𝒔 𝒐𝒇 𝒄𝒐𝒏𝒗𝒆𝒓𝒈𝒆𝒏𝒄𝒆 + ∞
𝑒𝑥𝑡𝑒𝑛𝑑 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 − 𝑅 𝑎𝑛𝑑 𝑅; 𝑤𝑒 𝑠𝑎𝑦 𝑡ℎ𝑒 𝑠𝑒𝑟𝑖𝑒𝑠 ℎ𝑎𝑠 𝒓𝒂𝒅𝒊𝒖𝒔𝒐𝒇 𝒄𝒐𝒏𝒗𝒆𝒓𝒈𝒆𝒏𝒄𝒆 𝑹
• Find the Interval of Convergence :
The usual procedure for finding the interval of convergence of a power series is to apply the ratio
test for absolute convergence.
o Find the interval of convergence and radius of convergence of the following series.

• Power Series in 𝒙 − 𝒙𝟎 ∶
If 𝑥0 is a constant, and if 𝑥 is replaced by 𝑥 − 𝑥0 in

then, we have

For example:

The first of these is a power series in 𝑥 − 1 and the second is a power series in 𝑥 + 3. Note that a
power series in 𝑥 is a power series in 𝑥 − 𝑥0 in which 𝑥0 = 0.
More generally, the Taylor series

is a power series in 𝑥 − 𝑥0 .
The set of values for which a power series in 𝑥 − 𝑥0 converges is always an interval centered at 𝑥 =
𝑥0, we call this the interval of convergence.
When the interval of convergence is :
𝑟𝑒𝑑𝑢𝑐𝑒𝑑 𝑡𝑜 𝑡ℎ𝑒 𝑠𝑖𝑛𝑔𝑙𝑒 𝑣𝑎𝑙𝑢𝑒 𝑥 = 𝑥0 : 𝑤𝑒 𝑠𝑎𝑦 𝑡ℎ𝑒 𝑠𝑒𝑟𝑖𝑒𝑠 ℎ𝑎𝑠 𝒓𝒂𝒅𝒊𝒖𝒔 𝒐𝒇 𝒄𝒐𝒏𝒗𝒆𝒓𝒈𝒆𝒏𝒄𝒆 𝑹 = 𝟎
{ 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒: 𝑤𝑒 𝑤𝑎𝑦 𝑡ℎ𝑒 𝑠𝑒𝑟𝑖𝑒𝑠 ℎ𝑎𝑠 𝒓𝒂𝒅𝒊𝒖𝒔 𝒐𝒇 𝒄𝒐𝒏𝒗𝒆𝒓𝒈𝒆𝒏𝒄𝒆 𝒂𝒕 𝑹 = +∞
𝑒𝑥𝑡𝑒𝑛𝑑𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑥0 − 𝑅 𝑎𝑛𝑑 𝑥0 + 𝑅: 𝑤𝑒 𝑠𝑎𝑦 𝑡ℎ𝑒 𝑠𝑒𝑟𝑖𝑒𝑠 ℎ𝑎𝑠 𝒓𝒂𝒅𝒊𝒖𝒔 𝒐𝒇 𝒄𝒐𝒏𝒗𝒆𝒓𝒈𝒆𝒏𝒄𝒆 𝑹

o Find the interval of convergence and radius of convergence of the series

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