Infinite Series Explained - Revision Notes

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12.4Infinite Series Explained

In these revision notes for Infinite Series Explained, we cover the following key points:

  • What are finite and infinite series? How do you identify them?
  • What are diverging and converging series?
  • How do we find the sum of all terms in converging geometric series?
  • What is the comparison test of convergence?
  • What are some special types of infinite series?
  • What is the ratio convergence test? When do we use it?
  • What is the root convergence test? When do we use it?

Infinite Series Explained Revision Notes

An infinite series is a series that has an infinite number of terms in it. Such series never terminate, i.e. they have no final value. An infinite series is either written in a summation notation:

S = n = 1xn

or by three dots after the last known term to indicate the continuation to infinity, i.e.

S = x1 + x2 + x3 + ⋯ + xn + ⋯

Some number series have a finite value that represents the despite having an infinite number of terms. They are called converging infinite series. For example, geometric series with a fractional common ratio smaller than 1 are infinite series, as the terms become smaller and smaller; the increase in sum becomes more and more irrelevant.

The condition for an infinite series to be convergent is that

n = 1xn = L

where L is a finite number (sometimes, we call it a limit value or simply, limit).

The common ratio R of converging infinite geometric series must be smaller than 1 when these series are converging. Extending this reasoning to also include negative ratios that are greater than -1, we obtain the condition for a geometric series to be converging:

-1 < R < 1

The formula for calculation of the infinite geometric series with a common ratio between -1 and 1 is

S = x1/1 - R

This is because in the formula of geometric series with |R| < 1

Sn = x1 ∙ (1 - Rn)/1 - R

Rn points towards zero when n points towards infinity.

The comparison test of convergence with two infinite series

n = 1xn and n = 1yn

says that if xn ≤ yn for sufficiently large values of n, then

  1. If n = 1yn converges, then n = 1xn converges too, and
  2. If n = 1xn diverges, then n = 1yn diverges too.

The above rule is true not only for geometric series but for all infinite series.

There are some special types of infinite series, which have many important applications in practice. Some of them include

  1. The p-series. This is a special type of series which has the form
    n - 11/np
    This series is special because it is convergent for p > 1 and divergent for 0 < p ≤ 1.
  2. Harmonic series. They represent a special case of the p-series where p = 1, i.e
    n - 11/np = n - 11/n1 = n - 11/n
    This series is divergent, as it is impossible to reach to a limit point, despite the decrease in the terms value with the increase of n.
  3. Euler Series This too, is a special series, which has the following form
    Euler Series = n = 01/n!
    We write the name of Euler Series (otherwise known as the Euler's Number) by e. It is worth to mention here that the counting of such a series elements start from 0, not from 1. In this way, we obtain
    e = n = 01/n! = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ⋯
    This is a converging series, as the non-constant ratio between two consecutive terms is always smaller than 1.

Ratio Convergence Test consists of calculating the limit of the ratio between two consecutive terms when the number of terms points to infinity. Thus, if Σan is a series with positive terms, where a may be a variable, a monomial or an entire algebraic expression and supposing that

limx → ∞an + 1/an = L

then, the Ratio Test of Convergence rule says:

  1. If L < 1, then Σan is a convergent series;
  2. If L > 1, then Σan is a divergent series; and
  3. If L = 1, then the convergence test of Σan is inconclusive, i.e. we need to use other methods to study its convergence.

Root Convergence Test is particularly important in cases when the ratio test cannot provide an answer regarding the convergence of an infinite series. Thus, if Σan is an infinite series with non-negative terms and supposing that

limn → ∞⁡(an )1/n = L

where L is a finite number (it represents the limit of the series), then

  1. If L < 0, then Σan is convergent;
  2. If L > 0, then Σan is divergent;
  3. If L = 1, then the test is conclusive, i.e. it cannot give us an exact answer about the convergence of the given series Σan.

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