Based on your screenshot, it seems that the question asks you to use the root test to determine the convergence of the series with the general term:

$\left( \frac{3n + 2}{9n + 1} \right)^{2n}$

You entered the following values:

• Limit calculation: $\frac{1}{3}$
• Comparison: "less than 1"
• Conclusion: "converges"

Let's break down the solution and check why some of your answers were marked incorrect:

1. Root Test: The root test states that for a series $\sum a_n$, if we consider $\lim_{n \to \infty} \sqrt[n]{|a_n|}$:

   • If this limit is less than 1, the series converges absolutely.
   • If this limit is greater than 1, the series diverges.
   • If this limit equals 1, the test is inconclusive.

2. Finding the Limit:

   • The term given is $\left( \frac{3n + 2}{9n + 1} \right)^{2n}$.
   • Taking the $n$th root, we have: $\sqrt[n]{\left( \frac{3n + 2}{9n + 1} \right)^{2n}} = \left( \frac{3n + 2}{9n + 1} \right)^2$
   • Now, calculate $\lim_{n \to \infty} \left( \frac{3n + 2}{9n + 1} \right)^2$:
     • Simplifying $\frac{3n + 2}{9n + 1}$ as $n \to \infty$, we get $\frac{3}{9} = \frac{1}{3}$.
     • Thus, $\lim_{n \to \infty} \left( \frac{3n + 2}{9n + 1} \right)^2 = \left( \frac{1}{3} \right)^2 = \frac{1}{9}$.

3. Conclusion:

   • Since $\frac{1}{9} < 1$, the series converges by the root test.

Corrections

• The correct limit is $\frac{1}{9}$ rather than $\frac{1}{3}$.
• Your conclusion that the series converges was correct, but updating the limit to $\frac{1}{9}$ should ensure full credit.

Would you like a detailed explanation of each step or further assistance?

Related Questions:

1. What other convergence tests could be used for series with terms raised to the power $2n$?
2. How does the ratio test compare with the root test for series with exponential terms?
3. Why does the limit approaching a value less than 1 imply convergence in the root test?
4. Could there be an error in the convergence if a term with $n$th power behaves differently?
5. When is the root test considered inconclusive?

Tip:

Always simplify the expression within the root to avoid calculation errors when applying the root test.