Infinite Series

Objectives

• Explain the meaning of the sum of an infinite series.
• Calculate the sum of a geometric series.

Sums and Series

• An infinite series is a sum of an infinite set of terms.
  • It is often written in the form:

• In computing an infinite series, we utilize the partial sum of an infinite series.
  • It means that instead of computing the sum of an infinite terms, we focus on computing the sum of the first terms instead.

Definition
Partial Sum

The partial sum of an infinite series, , is the sum of the first terms in a sequence .

• Using the partial sum of an infinite series, we can construct a sequence of partial sums , where the term of this sequence represents how many terms has been added.
• We can then observe the behavior of this sequence as we consider the limit as .
  • If the sequence of partial sums converges to a finite value, the infinite series itself converges.
  • If the sequence of partial sums diverges, then the infinite series diverges.

Definition
Infinite Series

An infinite series is an expression in the form:

For each positive integer , the infinite series forms a sequence of partial sums .
Each term in this sequence is defined as:

The value of the infinite series is the limit of the sequence of partial sums.

• If the sequence converges, then the infinite series converges.
• If the sequence diverges, the the infinite series diverges

Properties of Convergent Series

• Like finite series, some algebraic properties apply to convergent series as well.

Theorem
Properties of Convergent Series

Let and be convergent series. Then:

• Sum/Difference Rule: If converges, then:

• Constant Multiple Rule: Given any real number , if the series converges, then:

Harmonic Series

• The harmonic series is defined as the series:

• This series diverges, but it diverges very slowly.
• We can show this slow growth by showing terms of the its sequence of partial sums.

• At this point, it is still not clear whether or not this series diverges.
• We can, however, prove its diverges analytically.

Proof
Proving that the Harmonic Series Diverges

Let be the sequence of partial sums of the infinite series .
We begin by writing the first partial sums.

Notice that on the last two terms of

Therefore,

Using the same logic, we can see a pattern when we consider :

By continuing this pattern, we can see that for powers of for :

Knowing that:

And since , then, by comparison, the sequence is unbounded and thus, it diverges.

Geometric Series

• A geometric series is any series that we can write in the form:

• Just like in a geometric sequence:
  • represents the initial term
  • represents the common ratio
  • Each term in a geometric series is exactly times the previous term.
• In a geometric series, we are concerned when does a geometric series converges.
  • To do this, we can consider evaluating the partial sum of a geometric series for any given .

Proof
Convergence of a Geometric Series

Let be a sequence of partial sums of the geometric series in the form.

By definition of the partial sum, is given by

For this example, we assume that .
If , anything multiplied by is and the sequence and the series converges to .
We now look into when the limit of this sequence of partial sums converges

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Case 1:

When , the partial sum simplifies to:

• As , .
  • Therefore, the sequence of partial sums is unbounded and thus it diverges.
  • This means that the infinite series diverges for .

Case 2: and

See also: The Limit of a Geometric Sequence
When , the sequence becomes unbounded and thus diverges.
Therefore, the sequence of partial sums also diverges when or

When , the sequence is bounded but still diverges
Therefore, the sequence of partial sums diverges when
In either case, the infinite series diverges.

Case 3:

In these cases, we do some manipulation to analyze these cases instead.
We multiply the partial sum of a geometric series by .

Since , the sequence converges to . (Proof: + Case) (Proof: - Case)
And so:

Therefore, the series converges when .

Definition
Geometric Series

A geometric series is a series in the form of:

The convergence of the geometric series depends on the value of .
For any

• If , the series converges and

• If , the series diverges.

• Some geometric series appear in slightly different forms.
• For instance, consider the series:

• We can see that this is a geometric series by rewriting out the terms.

• By writing it this way, we see that and
  • Therefore, we can confidently say:

Rewriting Decimals with Geometric Series

• We can rewrite repeating decimals as fractions of integers using geometric series.
  • For instance, consider the decimal
  • By using powers of , we can say that this fraction can be written as:

• We can rewrite the decimal part as a fraction of a power of .

• Then, from the third term and onward, we can rewrite them as multiples of .

• Using the definition of the infinite geometric series, we identify that and .
  • Therefore:

• Since and , the series converges.
  • Therefore, we use the infinite geometric series formula to evaluate where it converges.

• Evaluating the infinite series from the previous expression, we get:

Proof
?

In popular media, the notion that is equal to may seem elusive and counterintuitive at first. On one side, it might seem reasonable and true. On the other hand, some people might say that this number might be equal to something else.

But before we break down this problem, let us see the motivation and the origin behind this problem.

Motivation

Consider the fraction .
Its decimal representation can be expressed as
Therefore:

Multiplying on both sides of the equation, we get:

Proof
We rewrite the repeating decimal as a geometric series representation.
First, we start by breaking down the decimal as a sum of powers of ten.

Then, we express each term as a fraction of powers of .

We can then express each term as a multiple of .

We can now rewrite this as a geometric series.
Let and .

Since , this geometric series is convergent. Therefore:

Telescoping Series

• A telescoping series is a series when most of the terms cancel in each of the partial sums, leaving only some of the first terms and some of the last terms.
• For example: consider the series

• Writing out its partial sums, we get:

• In general:

• Notice that as , as well.
  • Therefore, this series diverges.

Definition
Telescoping Series

A telescoping series is when most terms in a series cancel in each of the partial sums, leaving only some of the first terms and some of the last terms.

It has the general form:

And the partial sum of this series is:

If the sequence converges to some finite number , then this telescoping series converges to: