---
title: "State and prove final value theorem"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:final-value-theorem-prove"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/final-value-theorem-prove
site: BCT Engineering
---

# State and prove final value theorem

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2073 Magh, 1+4 marks; 2072 Asoj, 5 marks; 2071 Bhadra; 2069 Poush; 2068 Magh, 5 marks (asked 5 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/final-value-theorem-prove
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

State and prove final value theorem for z-transform.

## Short answer

If x(k) tends to a finite limit and (1–z⁻¹)X(z) has all poles inside |z|=1, then x(∞)=lim_{z→1}(z–1)X(z). Proof: Use Z[x(k+1)–x(k)]=(z–1)X(z)–z x(0), take z→1, and evaluate the telescoping sum yielding x(∞)–x(0).

## Answer

#### Statement
If X(z) = Z[x(k)] and (1 − z⁻¹)X(z) has all its poles inside the unit circle |z| = 1 (so that x(k) tends to a finite limit), then

x(∞) = lim (k → ∞) x(k) = lim (z → 1) (z − 1) X(z) = lim (z → 1) (1 − z⁻¹) X(z)

#### Proof
Use the shifting property Z[x(k + 1)] = z X(z) − z x(0). Then

Z[x(k + 1) − x(k)] = z X(z) − z x(0) − X(z) = (z − 1) X(z) − z x(0)

By definition of the z-transform,

Σ (k = 0 to ∞) [x(k + 1) − x(k)] z⁻ᵏ = (z − 1) X(z) − z x(0)

Take the limit z → 1 on both sides. The left side becomes a telescoping sum:

lim (n → ∞) Σ (k = 0 to n) [x(k + 1) − x(k)]
= lim (n → ∞) [x(n + 1) − x(0)] = x(∞) − x(0)

The right side becomes lim (z → 1) (z − 1) X(z) − x(0).

Hence

x(∞) − x(0) = lim (z → 1) (z − 1) X(z) − x(0)

so

**x(∞) = lim (z → 1) (z − 1) X(z)**

Condition: the theorem is valid only if x(∞) exists, i.e. all poles of (z − 1)X(z) lie inside the unit circle.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the final value theorem?
   A. x(∞) = lim_{z→0} X(z)
   B. x(∞) = lim_{z→∞} X(z)
   C. x(∞) = lim_{z→1} (z-1) X(z)
   D. x(∞) = lim_{z→1} X(z)
   Answer: C. (z-1)X(z) as z→1. (Key point: Statement)

2. Which property starts the proof?
   A. Z[x(k+1)] = X(z)/z
   B. Z[x(k+1)] = zX(z)
   C. Z[x(k-1)] = zX(z)
   D. Z[x(k+1)] = zX(z) - z x(0)
   Answer: D. Subtracting X(z) gives (z-1)X(z) - z x(0). (Key point: Shift property)

3. Why does the left side equal x(∞) - x(0)?
   A. The sum Σ[x(k+1)-x(k)] telescopes to x(n+1)-x(0)
   B. Terms alternate and cancel in pairs
   C. z⁻ᵏ → 0
   D. x is constant
   Answer: A. Partial sums collapse. (Key point: Telescoping)

4. When is the theorem valid?
   A. Only when x(0) = 0
   B. x(∞) exists, i.e. poles of (z-1)X(z) are inside the unit circle
   C. Only for finite-length sequences
   D. Always
   Answer: B. Otherwise the limit does not exist. (Key point: Condition)

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