Math Problem Statement

Determine whether the series \(\sum_{n=1}^{\infty} \frac{3^n + 4^n}{6^n}\) converges or diverges. If it converges, find its sum.

Solution

Let's walk through the problem to determine if the given series converges or diverges, and if it converges, find the sum.

Given:

n=13n+4n6n\sum_{n=1}^{\infty} \frac{3^n + 4^n}{6^n}

Step 1: Rewrite the series

The term inside the summation can be split: 3n+4n6n=3n6n+4n6n=(36)n+(46)n\frac{3^n + 4^n}{6^n} = \frac{3^n}{6^n} + \frac{4^n}{6^n} = \left(\frac{3}{6}\right)^n + \left(\frac{4}{6}\right)^n

This simplifies to: (12)n+(23)n\left(\frac{1}{2}\right)^n + \left(\frac{2}{3}\right)^n

Step 2: Analyze the two series

Both components are geometric series:

  • For (12)n\left(\frac{1}{2}\right)^n:

    • This is a geometric series with r=12r = \frac{1}{2}. Since r<1|r| < 1, it converges.
    • Sum of the series: 1/211/2=1\frac{1/2}{1 - 1/2} = 1
  • For (23)n\left(\frac{2}{3}\right)^n:

    • This is another geometric series with r=23r = \frac{2}{3}. Since r<1|r| < 1, it also converges.
    • Sum of the series: 2/312/3=2\frac{2/3}{1 - 2/3} = 2

Step 3: Find the total sum

Since both geometric series converge, the total sum is the sum of their individual sums: 1+2=31 + 2 = 3

Step 4: Conclusion

The given series converges and its sum is 33.


Correct Answer:

The correct choice is A:

  • "The series converges because it is the sum of two geometric series, each with r<1|r| < 1."
  • The sum of the series is: 33

Do you have any questions, or would you like further clarification?


Here are five related questions for further exploration:

  1. What are the conditions for a geometric series to converge?
  2. How would the sum change if one of the terms had r1|r| \geq 1?
  3. Can you generalize the sum of a series like n=1an+bncn\sum_{n=1}^{\infty} \frac{a^n + b^n}{c^n}?
  4. What is the relationship between convergence tests like the Ratio Test and geometric series?
  5. How does changing the base of the exponents (e.g., 5n5^n or 7n7^n) affect convergence?

Tip: Always check if you can split complex series into simpler parts, such as sums of geometric series, to analyze convergence easily.

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Math Problem Analysis

Mathematical Concepts

Series
Geometric Series
Convergence
Summation

Formulas

Geometric series formula: \(\sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r}\) for |r| < 1

Theorems

Geometric Series Convergence Theorem

Suitable Grade Level

College-level mathematics or advanced high school