https://r2.erweima.ai/mathbot/image/1721929499402125bd5ot.jpg

To solve the given problem, we need to evaluate the following expression to three decimal places:

\[ \sum_{r=2}^{\infty} 3.2^{1-r} + \sum_{r=2}^{12} 3.2^{1-r} \]

Let's break this down into two parts:

### Part 1: Infinite Sum

The first part is an infinite geometric series:

\[ \sum_{r=2}^{\infty} 3.2^{1-r} \]

The general term of this series is \( 3.2^{1-r} \). To find the sum of this infinite geometric series, we identify the first term \(a\) and the common ratio \(r\).

The first term \(a\) when \(r=2\) is:

\[ a = 3.2^{1-2} = 3.2^{-1} = \frac{1}{3.2} \]

The common ratio \(r\) is:

\[ r = 3.2^{-1} = \frac{1}{3.2} \]

The sum \(S\) of an infinite geometric series \( \sum_{n=0}^{\infty} ar^n \) is given by:

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

In our case:

\[ S = \frac{\frac{1}{3.2}}{1 - \frac{1}{3.2}} = \frac{\frac{1}{3.2}}{\frac{2.2}{3.2}} = \frac{1}{2.2} \approx 0.455 \]

### Part 2: Finite Sum

The second part is a finite geometric series from \(r=2\) to \(r=12\):

\[ \sum_{r=2}^{12} 3.2^{1-r} \]

The sum \(S\) of the first \(n\) terms of a geometric series \( \sum_{k=0}^{n-1} ar^k \) is given by:

\[ S_n = a \frac{1-r^n}{1-r} \]

Here, \(a = 3.2^{-1} = \frac{1}{3.2}\), the common ratio \(r = \frac{1}{3.2}\), and \(n = 11\) (since \(12 - 2 + 1 = 11\)).

\[ S_{11} = \frac{1}{3.2} \cdot \frac{1 - \left(\frac{1}{3.2}\right)^{11}}{1 - \frac{1}{3.2}} = \frac{1}{3.2} \cdot \frac{1 - \left(\frac{1}{3.2}\right)^{11}}{\frac{2.2}{3.2}} = \frac{1}{2.2} \left(1 - \left(\frac{1}{3.2}\right)^{11}\right) \]

Calculating \( \left(\frac{1}{3.2}\right)^{11} \):

\[ \left(\frac{1}{3.2}\right)^{11} \approx 2.14 \times 10^{-6} \]

Thus:

\[ S_{11} \approx \frac{1}{2.2} \left(1 - 2.14 \times 10^{-6}\right) \approx 0.455 - 9.73 \times 10^{-7} \approx 0.455 \]

### Total Sum

Now we add the two parts together:

\[ 0.455 + 0.455 = 0.910 \]

Therefore, the value of the given expression to three decimal places is:

\[ \boxed{0.910} \]

Do you want details or have any questions?

Here are 5 questions you might ask next:
1. How do I solve other types of infinite series?
2. Can you explain the properties of geometric series?
3. How do you determine the convergence of a series?
4. What is the difference between finite and infinite series?
5. Can you help me solve a different math problem?

**Tip:** When working with geometric series, always check if the series converges (i.e., if the common ratio's absolute value is less than 1) before attempting to find the sum.

Learn how to solve a complex mathematical problem involving the summation of infinite and finite geometric series. This problem explores the concepts of infinite series convergence, common ratio calculation, and series summation techniques. Suitable for advanced mathematics students and enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation