How would I complete the attached question in Python using only Numpy, recursion loops, or basic math functions? ### Question 2Prove that\[\frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \frac{1}{256} + \cdots = \frac{1}{3}\]This mathematical problem asks students to demonstrate that the sum of the infinite series, starting with \(\frac{1}{4}\) and with each subsequent term being the previous term divided by 4, converges to \(\frac{1}{3}\).The series given in the problem is a geometric series where the first term \(a = \frac{1}{4}\) and the common ratio \(r = \frac{1}{4}\). The formula for the sum \(S\) of an infinite geometric series is:\[S = \frac{a}{1 - r}\]By substituting the given values:\[S = \frac{\frac{1}{4}}{1 - \frac{1}{4}} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3}\]Therefore, the series converges to \(\frac{1}{3}\).

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Database System Concepts

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ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

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How would I complete the attached question in Python using only Numpy, recursion loops, or basic math functions?

### Question 2

Prove that

\[
\frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \frac{1}{256} + \cdots = \frac{1}{3}
\]

This mathematical problem asks students to demonstrate that the sum of the infinite series, starting with \(\frac{1}{4}\) and with each subsequent term being the previous term divided by 4, converges to \(\frac{1}{3}\).

The series given in the problem is a geometric series where the first term \(a = \frac{1}{4}\) and the common ratio \(r = \frac{1}{4}\). The formula for the sum \(S\) of an infinite geometric series is:

\[
S = \frac{a}{1 - r}
\]

By substituting the given values:

\[
S = \frac{\frac{1}{4}}{1 - \frac{1}{4}} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3}
\]

Therefore, the series converges to \(\frac{1}{3}\).

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