What test are we using for # 30? The original problem is written on scrap paper in pencil ### Sequence Convergence Analysis#### Example 10.2.28The sequence converges because $0.003 < 1$, so the sequence converges to zero.#### Example 10.2.29 Given that both $0.5 < 1$ and $0.75 < 1$, the sequence converges to $0 + 3 \cdot 0 = 0$.#### Example 10.2.30To evaluate the limit:\[\lim_{n \to \infty} \frac{e^n + \pi^n}{e^n} = \lim_{n \to \infty} \left(1 + \left( \frac{\pi}{e} \right)^n \right) = 1 + \lim_{n \to \infty} \left( \frac{\pi}{e} \right)^n\]This limit doesn't exist because $\left| \frac{\pi}{e} \right| > 1$. The image contains a mathematical expression written as a fraction. **Expression:**The numerator is $ e^{n + n} $.The denominator is $ e^n $.**Overall Expression:**\[\frac{e^{n+n}}{e^n}\]**Explanation:**This expression simplifies the exponent rules. According to the laws of exponents, when you divide similar bases, you subtract the exponents. Therefore, this expression can be simplified as follows:\[\frac{e^{n+n}}{e^n} = e^{n+n-n} = e^n\] This results in $ e^n $ due to the simplification of the exponents.

If x>1 then limn→∞xn does not exists.

because 0< < 1, the sequence converges monotonically to zero. 10.2.28 -0.003 < 1, so the sequence converges to zero. 10.2.29 Because |0.5| <1 and |0.75 < 1, the sequence converges to 0+3.0=0. en + " (:)") 10.2.30 lim lim (1+ which doesn't exist, because =1+ lim >1. > 1. 1. en n-00

because 0< < 1, the sequence converges monotonically to zero. 10.2.28 -0.003 < 1, so the sequence converges to zero. 10.2.29 Because |0.5| <1 and |0.75 < 1, the sequence converges to 0+3.0=0. en + " (:)") 10.2.30 lim lim (1+ which doesn't exist, because =1+ lim >1. > 1. 1. en n-00

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

What test are we using for # 30? The original problem is written on scrap paper in pencil

### Sequence Convergence Analysis

#### Example 10.2.28
The sequence converges because $0.003 < 1$, so the sequence converges to zero.

#### Example 10.2.29
Given that both $0.5 < 1$ and $0.75 < 1$, the sequence converges to $0 + 3 \cdot 0 = 0$.

#### Example 10.2.30
To evaluate the limit:
\[
\lim_{n \to \infty} \frac{e^n + \pi^n}{e^n} = \lim_{n \to \infty} \left(1 + \left( \frac{\pi}{e} \right)^n \right) = 1 + \lim_{n \to \infty} \left( \frac{\pi}{e} \right)^n
\]
This limit doesn't exist because $\left| \frac{\pi}{e} \right| > 1$.

The image contains a mathematical expression written as a fraction.

**Expression:**

The numerator is $ e^{n + n} $.
The denominator is $ e^n $.

**Overall Expression:**

\[
\frac{e^{n+n}}{e^n}
\]

**Explanation:**

This expression simplifies the exponent rules. According to the laws of exponents, when you divide similar bases, you subtract the exponents. Therefore, this expression can be simplified as follows:

\[
\frac{e^{n+n}}{e^n} = e^{n+n-n} = e^n
\]

This results in $ e^n $ due to the simplification of the exponents.

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