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.
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
Section: Chapter Questions
Problem 1RQ
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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 \(c^n \rightarrow 0\).**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F89f06fcb-a461-4580-b866-76a69e450ea3%2F44cd01fb-21e3-4624-9fd4-4ab5a415dfaf%2F03uwfz8_processed.jpeg&w=3840&q=75)
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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