3. Let c be a number with c < 1 Show that c can be expressed as el = 1 where d > 0. Then use the Binomial Formula to show that c≤nd ≤ for every index n 4. Use the above problem to prove that if c < 1, then c→ 0.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**3. Let \( c \) be a number with \(|c| < 1\). Show that \(|c|\) can be expressed as \(|c| = \frac{1}{1+d}\) where \( d > 0 \). Then use the Binomial Formula to show that**

\[
|c^n| \leq \frac{1}{1+nd} \leq \frac{1}{dn} \quad \text{for every index } n
\]

**4. Use the above problem to prove that if \(|c| < 1\), then \(c^n \rightarrow 0\).**
Transcribed Image Text:**3. Let \( c \) be a number with \(|c| < 1\). Show that \(|c|\) can be expressed as \(|c| = \frac{1}{1+d}\) where \( d > 0 \). Then use the Binomial Formula to show that** \[ |c^n| \leq \frac{1}{1+nd} \leq \frac{1}{dn} \quad \text{for every index } n \] **4. Use the above problem to prove that if \(|c| < 1\), then \(c^n \rightarrow 0\).**
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