Last modified: March 28, 2024

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Sequences and Series

Sequences

A sequence is an ordered list of numbers that can be viewed as a function mapping each natural number n to a specific value an. More formally, a sequence an is a function whose domain is the set of natural numbers, and the values are called the terms of the sequence:

a1,a2,a3,…,an,…

A sequence can either approach a particular value, in which case it is said to converge, or it can increase or oscillate indefinitely, in which case it diverges.

A sequence an converges to a limit a if, as n becomes very large, the values of an get arbitrarily close to a. Mathematically, this is expressed as:

limnβ†’βˆžan=a

This means for any small number Ο΅>0, there exists a large number N such that for all n>N, the distance between an and a is smaller than Ο΅. If no such limit exists, the sequence diverges.

Examples of Sequences

  1. Convergent Sequence:

Consider the sequence an=nn+2. This gives the sequence:

13,24,35,…

As nβ†’βˆž, the terms of the sequence approach 1:

limnβ†’βˆžnn+2=1

Therefore, this sequence converges to 1.

  1. Divergent Sequence:

Now consider the sequence an=4n. This gives the sequence:

4,16,64,…

As nβ†’βˆž, the terms of the sequence grow without bound, meaning this sequence diverges.

  1. Another Divergent Sequence:

Another example of a divergent sequence is an=n+1, which gives:

2,3,4,…

As nβ†’βˆž, the sequence grows indefinitely, and hence it diverges.

  1. Convergent Sequence:

Consider the sequence an=1n3, which gives:

1,18,127,…

As nβ†’βˆž, the terms of this sequence get smaller and smaller, approaching 0:

limnβ†’βˆž1n3=0

Hence, this sequence converges to 0.

Partial Sums

The partial sum sn of a sequence an is the sum of the first n terms of the sequence:

sn=a1+a2+β‹―+an

Partial sums are used to analyze the behavior of series, where the idea is to observe whether the sum of the terms converges to a specific value as more terms are added. For example, consider the following partial sums:

A series formed by a sequence is said to converge if the sequence of partial sums converges as nβ†’βˆž.

Series

A series is the sum of the terms of a sequence. If the sequence of partial sums sn converges to a limit s, the series is said to converge to that limit s. Mathematically, the infinite series is written as:

βˆ‘k=1∞ak=limnβ†’βˆžsn=limnβ†’βˆž(a1+a2+β‹―+an)=s

If the partial sums do not approach a finite limit as n increases, then the series is said to be divergent.

Geometric Series and Rational Functions

A geometric series is a series of the form:

βˆ‘k=0∞rk=11βˆ’r,for |r|<1

This result can be used to represent certain rational functions as infinite series. For example:

11βˆ’x=βˆ‘k=0∞xk,for |x|<1

This is useful in time series analysis when dealing with autoregressive models and other representations involving rational functions.

Examples of Convergent Series

I. Geometric Series:

A geometric series is a series where each term is a constant multiple (called the common ratio) of the previous term. A simple example is the following geometric series:

βˆ‘k=0∞13k

This series converges to:

βˆ‘k=0∞13k=11βˆ’13=32

II. P-Series (convergent for p>1):

A p-series is of the form:

βˆ‘k=1∞1kp

For p=2, the series converges to:

βˆ‘k=1∞1k2=Ο€26

III. Alternating Series:

An alternating series is a series where the signs of the terms alternate between positive and negative. A famous example is:

βˆ‘k=1∞(βˆ’1)k+1k

This series converges to:

βˆ‘k=1∞(βˆ’1)k+1k=ln⁑(2)

Examples of Divergent Series

I. Geometric Series with growth factor greater than 1:

Consider the geometric series:

βˆ‘k=1∞4k=4+16+64+…

Since the ratio between successive terms is greater than 1, this series diverges as the partial sums grow without bound.

II. Arithmetic Series:

In an arithmetic series, the difference between successive terms is constant. For example:

βˆ‘k=1∞(2k+3)=5+7+9+…

The terms grow linearly, and the series diverges as nβ†’βˆž.

III. Harmonic Series:

The harmonic series is:

βˆ‘k=1∞1k=1+12+13+…

Although the terms 1k approach 0 as kβ†’βˆž, the sum of the terms grows without bound. Hence, the harmonic series diverges.

Absolute Convergence

A series is absolutely convergent if the series of the absolute values of its terms is convergent:

βˆ‘k=1∞|ak|

Absolute convergence implies convergence. This is a stronger condition than regular convergence, as a series can converge without being absolutely convergent (for example, alternating series).

Convergence Tests

When dealing with infinite series, it’s important to determine whether the series converges (approaches a finite value) or diverges (grows without bound or oscillates without settling). To do this, mathematicians use various convergence tests, each suited for different types of series. Below are some of the most commonly used tests for determining whether an infinite series converges or diverges:

I. Integral Test:

The integral test compares a series to the integral of a continuous, positive, decreasing function. Suppose an is a sequence and f(x) is a continuous, positive, decreasing function such that f(n)=an for all nβ‰₯1. If the integral of f(x) from 1 to infinity converges, then the series converges. Otherwise, the series diverges.

βˆ‘n=1∞anconverges if and only if∫1∞f(x)dxconverges.

Example: Consider the series βˆ‘n=1∞1n2. Compare this to the integral ∫1∞1x2dx, which converges to 1. Hence, the series converges.

II. Comparison Test:

The comparison test compares the given series to a known convergent or divergent series. If the terms of the series an are smaller than the terms of a known convergent series bn, and all terms are positive, then the series βˆ‘an also converges. Similarly, if the terms are larger than those of a known divergent series, the series βˆ‘an diverges.

0≀an≀bnandβˆ‘bnconvergesβ‡’βˆ‘anconverges.

Example: To check if βˆ‘n=1∞1n3+2 converges, compare it to βˆ‘n=1∞1n3, which is a convergent p-series with p>1. Since the terms of 1n3+2 are smaller, the series converges by comparison.

III. Limit Comparison Test:

The limit comparison test is useful when the terms of the series are not directly comparable to a known series but have similar behavior as nβ†’βˆž. If the limit of the ratio between the terms of two series is a finite, positive constant, both series either converge or diverge together.

limnβ†’βˆžanbn=cwhere0<c<∞.

Example: Consider the series βˆ‘n=1∞n2+12n2+3. Compare it to βˆ‘n=1∞1n2. The limit of the ratio of terms is:

limnβ†’βˆžn2+12n2+31n2=12.

Since the comparison series βˆ‘1n2 converges, the original series also converges.

IV. Alternating Series Test (Leibniz's Test):

This test is used for series where the terms alternate in sign. For an alternating series βˆ‘(βˆ’1)nan, if the absolute value of the terms an decreases monotonically (i.e., an+1≀an) and limnβ†’βˆžan=0, then the series converges.

Example: Consider the alternating harmonic series βˆ‘n=1∞(βˆ’1)n+1n, which is of the form (βˆ’1)n+11n. Since 1n decreases and approaches 0 as nβ†’βˆž, the series converges.

V. Ratio Test:

The ratio test is particularly useful for series involving factorials or exponential terms. To apply the ratio test, compute the limit of the absolute value of the ratio of successive terms:

L=limnβ†’βˆž|an+1an|.

Example: For the series βˆ‘n=1∞n!nn, applying the ratio test gives:

L=limnβ†’βˆž|(n+1)!/(n+1)n+1n!/nn|=limnβ†’βˆž(n+1)n+1β‹…(nn+1)n=0.

Since L=0, the series converges.

VI. Root Test:

The root test, or Cauchy's root test, involves taking the n-th root of the absolute value of the terms of the series. Let:

L=limnβ†’βˆž|an|n.

Example: For the series βˆ‘n=1∞(34)n, applying the root test gives:

L=limnβ†’βˆž|34|nn=34.

Since L=34<1, the series converges.

Mean-Square Convergence

In the context of stochastic processes, mean-square convergence is an important concept for analyzing the behavior of sequences of random variables. Suppose X1,X2,X3,… is a sequence of random variables representing a stochastic process. We say that Xn converges to a random variable X in the mean-square sense if the expected value of the squared difference between Xn and X approaches zero as n becomes large. Formally, this is expressed as:

E[(Xnβˆ’X)2]β†’0asnβ†’βˆž

This definition means that as nβ†’βˆž, the random variables Xn become increasingly close to X in the sense of the expected value of their squared differences, providing a measure of convergence in a probabilistic sense.

Inverting the MA(1) Model

Consider the MA(1) (Moving Average of order 1) model, which is commonly used in time series analysis. In this model, the process Xt at time t is defined as:

Xt=Zt+Ξ²Ztβˆ’1

where Zt represents white noise, which is a sequence of independent, identically distributed random variables with zero mean and constant variance ΟƒZ2. The parameter Ξ² is a constant that defines the relationship between the current observation and the previous white noise term.

We are interested in expressing Zt as an infinite sum involving the past values of Xt. Through algebraic manipulation, we can express Zt as:

Zt=βˆ‘k=0∞(βˆ’Ξ²)kXtβˆ’k

This infinite sum can be shown to converge in the mean-square sense, provided certain conditions on Ξ² are satisfied. Specifically, we need to ensure that the sum of these terms remains bounded as nβ†’βˆž, which ensures that the approximation βˆ‘k=0n(βˆ’Ξ²)kXtβˆ’k becomes increasingly close to Zt.

Autocovariance Function of the MA(1) Process

The autocovariance function Ξ³(k) of a stochastic process measures the covariance between Xt and Xt+k for different time lags k. For the MA(1) process, the autocovariance function is relatively simple due to the limited memory of the process, which only depends on the current and previous white noise terms.

The autocovariance function for the MA(1) process is given by:

Ξ³(k)={(1+Ξ²2)ΟƒZ2,if k=0Ξ²ΟƒZ2,if k=10,if k>1

For negative values of k, the autocovariance function is symmetric, meaning that Ξ³(βˆ’k)=Ξ³(k).

This shows that the MA(1) process has non-zero autocovariances only for the first lag (i.e., between Xt and Xtβˆ’1), and beyond that, the covariance is zero due to the white noise properties of Zt.

Series Convergence in Mean-Square Sense

To investigate the convergence of the infinite series expression for Zt, consider the partial sum of the first n terms:

βˆ‘k=0n(βˆ’Ξ²)kXtβˆ’k

We want to ensure that this sum converges to Zt in the mean-square sense. In other words, we need:

E[(βˆ‘k=0n(βˆ’Ξ²)kXtβˆ’kβˆ’Zt)2]β†’0asnβ†’βˆž

This expression represents the expected value of the squared difference between the partial sum and the true value of Zt, and we aim to show that this difference diminishes as more terms are added to the sum.

Expanding this expression:

E[(βˆ‘k=0n(βˆ’Ξ²)kXtβˆ’k)2βˆ’2E(βˆ‘k=0n(βˆ’Ξ²)kXtβˆ’kZt)+E[Zt2]]

Breaking this down into individual terms: - The first term involves the expected value of the squared partial sum: E[βˆ‘k=0nΞ²2kXtβˆ’k2]. - The second term represents the cross terms between the partial sum and Zt. - The third term is the variance of Zt, which is ΟƒZ2.

To achieve mean-square convergence, the total expression must approach 0 as nβ†’βˆž.

Condition for Convergence

For the series to converge in the mean-square sense, we require that the individual terms involving powers of Ξ² decay sufficiently fast. Specifically, we need:

ΟƒZ2Ξ²2(n+2)β†’0asnβ†’βˆž

This will only occur if:

|Ξ²|<1

When |Ξ²|<1, the powers of Ξ² diminish as n increases, ensuring that the sum remains bounded and that the partial sums converge to Zt.

Invertibility Condition

The condition |Ξ²|<1 is also known as the invertibility condition for the MA(1) process. This condition guarantees that the moving average process can be expressed as an infinite autoregressive (AR) process. In terms of the polynomial Ξ²(B)=1+Ξ²B (where B is the backshift operator), the invertibility condition states that the root of the polynomial must lie outside the unit circle in the complex plane, meaning:

|Ξ²|<1

Thus, the invertibility condition ensures that the MA(1) process can be uniquely represented as an AR(∞) process, which is crucial for the identification and estimation of time series models.

Table of Contents

    Sequences and Series
    1. Sequences
      1. Examples of Sequences
    2. Partial Sums
    3. Series
    4. Geometric Series and Rational Functions
    5. Examples of Convergent Series
    6. Examples of Divergent Series
    7. Absolute Convergence
    8. Convergence Tests
    9. Mean-Square Convergence
      1. Inverting the MA(1) Model
      2. Autocovariance Function of the MA(1) Process
      3. Series Convergence in Mean-Square Sense
      4. Condition for Convergence
      5. Invertibility Condition