https://r2.erweima.ai/mathbot/image/1723683778528sozuwlor.jpg

The equation in the image is:

\[
\frac{1}{2^{n+1}} + \frac{1}{2^{n+2}} + \frac{1}{2^{n+3}} + \cdots
\]

This is an infinite geometric series where each term after the first is a constant multiple (common ratio) of the previous term. The first term of the series is \( \frac{1}{2^{n+1}} \), and the common ratio \( r \) is \( \frac{1}{2} \).

To find the sum \( S \) of an infinite geometric series with the first term \( a_1 \) and common ratio \( r \) (where \( |r| < 1 \)), the formula is:

\[
S = \frac{a_1}{1 - r}
\]

Substituting the values:

\[
S = \frac{\frac{1}{2^{n+1}}}{1 - \frac{1}{2}}
\]

Simplifying the denominator:

\[
S = \frac{\frac{1}{2^{n+1}}}{\frac{1}{2}} = \frac{1}{2^{n+1}} \times 2 = \frac{2}{2^{n+1}} = \frac{1}{2^n}
\]

Thus, the sum of the series is:

\[
\boxed{\frac{1}{2^n}}
\]

Would you like further details or explanations on this solution? Here are some related questions you might find helpful:

1. How is the sum of an infinite geometric series derived?
2. What conditions must be met for an infinite geometric series to converge?
3. Can this method be applied to any geometric series, and if not, why?
4. How does the common ratio affect the convergence of the series?
5. What are some practical applications of infinite geometric series in real life?
6. How would the sum change if the common ratio were different?
7. What is the difference between finite and infinite geometric series?
8. Can you provide another example where this formula is used?

**Tip:** When working with series, always check whether it is convergent or divergent before attempting to find the sum.

Explore the solution to finding the sum of an infinite geometric series step-by-step. This problem focuses on the formula for the sum of an infinite geometric series, using specific values and demonstrating simplification techniques. Suitable for advanced high school or college-level students studying series and sequences in mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation