# Week 3

## Series and Series Representation
- Objectives
- Recall infinite series and their convergence
- Examine geometric series
- Represent rational functions as a geometric series

- Sequence and series
- Sequence: a list of numbers in definite order
- Partial sums: Sn = a1 + a2 +... + an
- 收斂：對於級數，若當 n 趨近於正無窮大時 Sn 趨向一個有限的極限

- Geometric series (幾何級數、等比數列)
- 收斂定義：乘數的絕對值小於1
- What we've learned
- The definition of infinite series and their convergence
- Geometric series is convergent if the multiplier has norm less than 1
- how to represent some rational functions as a geometric series

## Backward shift operator
- Objectives
- Define and utilize backward shift operator

- Backward shift operator: BXt = Xt-1 (e.g., random walk)

- What we've learned
- The definition of the Backward shift operator
- How to utilize backward shift operator to write MA(q) and AR(p) processes

## Introduction to Invertibility
- Objectives
- Learn invertibility of a stochastic process
- Whar we've learned
- Definition of invertibility of a sochastic process
- Invertibility condition guarantees unique MA process corresponding to observer ACF

## Durity
- Objectives
- invertibility condition for MA(q) processes
- Discover stationarity condition for AR(p) processes
- Relate MA and AR processes through duality
- What we've learned
- Invertibility condition for MA(q) processes
- Stationarity condition for AR(p) processes
- Duality MA and AR processes

## Mean Square Convergence (Optional)
- Objectives
- Learn mean-square convergence
- Formulate necessary and sufficient condition for invertibility of MA(1)