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# System prepended metadata

title: Week 3

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# 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)

