Python How do I adjust the loop for the left hand side of the equation to correctly calculate the problem in the photo?
Python How do I adjust the loop for the left hand side of the equation to correctly calculate the problem in the photo?
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Python
How do I adjust the loop for the left hand side of the equation to correctly calculate the problem in the photo?
![**Proof of Natural Logarithm Convergence Using Python**
In this educational example, we aim to prove the following mathematical concept using Python:
\[ \frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots = \ln(2) \]
where \(\ln\) stands for the natural logarithm.
**Python Code Explanation**
1. **Importing Libraries**
```python
import numpy as np
```
We import the `numpy` library as `np` to utilize its functions for numerical calculations, including the natural logarithm.
2. **Setting Initial Values**
```python
N = 4
LHSQ2 = 0
```
Here, we set \( N = 4 \) initially, but you should choose a value larger than 25 to achieve better convergence. `LHSQ2` is initialized to 0 to store the sum of the series.
3. **Calculating the Series**
```python
for i in range(1, N + 1):
LHSQ2 = LHSQ2 - (1 / i) + (1 / (-1 * i))
```
Using a `for` loop, we iterate from 1 to \( N \) (inclusive) to calculate the left-hand side of the equation. We accumulate the alternating series in `LHSQ2` by adding and subtracting reciprocals according to the series pattern.
4. **Printing Left-Hand Side Result**
```python
print('The left hand side equals', LHSQ2, '.')
```
This outputs the computed value of the series.
5. **Calculating the Right-Hand Side**
```python
RHSQ2 = np.log(2)
print('The right hand side equals', RHSQ2, ".")
```
We compute the natural logarithm of 2 using `np.log(2)` and print the result. This serves as the reference value to which the series should converge.
By running this code with \( N \) larger than 25, you can observe that the series increasingly approaches \(\ln(2)\), demonstrating the convergence of the series to the natural logarithm of 2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20ef5b89-bdf5-4ebf-bc1c-34f412b810c9%2F6692f3c4-ad5a-40a2-b6ad-f2bf67e5269e%2Fr4kwgf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Proof of Natural Logarithm Convergence Using Python**
In this educational example, we aim to prove the following mathematical concept using Python:
\[ \frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots = \ln(2) \]
where \(\ln\) stands for the natural logarithm.
**Python Code Explanation**
1. **Importing Libraries**
```python
import numpy as np
```
We import the `numpy` library as `np` to utilize its functions for numerical calculations, including the natural logarithm.
2. **Setting Initial Values**
```python
N = 4
LHSQ2 = 0
```
Here, we set \( N = 4 \) initially, but you should choose a value larger than 25 to achieve better convergence. `LHSQ2` is initialized to 0 to store the sum of the series.
3. **Calculating the Series**
```python
for i in range(1, N + 1):
LHSQ2 = LHSQ2 - (1 / i) + (1 / (-1 * i))
```
Using a `for` loop, we iterate from 1 to \( N \) (inclusive) to calculate the left-hand side of the equation. We accumulate the alternating series in `LHSQ2` by adding and subtracting reciprocals according to the series pattern.
4. **Printing Left-Hand Side Result**
```python
print('The left hand side equals', LHSQ2, '.')
```
This outputs the computed value of the series.
5. **Calculating the Right-Hand Side**
```python
RHSQ2 = np.log(2)
print('The right hand side equals', RHSQ2, ".")
```
We compute the natural logarithm of 2 using `np.log(2)` and print the result. This serves as the reference value to which the series should converge.
By running this code with \( N \) larger than 25, you can observe that the series increasingly approaches \(\ln(2)\), demonstrating the convergence of the series to the natural logarithm of 2.
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