Welcome back to class! As we wrap up our study of Infinite Sequences and Series, it is time to put your knowledge to the test. This Chapter 11 Practice Test is designed to help you synthesize the various convergence tests and power series manipulations we have covered.
Focus Areas
For this specific practice session, please focus your attention on Questions #1 through #13. These problems cover the core methodologies you will need for the exam, including:
- Convergence & Divergence Tests: You will need to determine which test applies to specific series, such as the Ratio Test, Root Test, Integral Test, and Comparison Tests.
- Radius & Interval of Convergence: Solving for $R$ and the interval $I$ for power series.
- Power Series Representations: manipulating geometric series to represent functions like $f(x) = \frac{x}{(1+4x)^2}$.
Key Formulas to Review
Before diving into the problems, recall the conditions for the Ratio Test, which is often crucial for power series:
$$ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| $$
- If $L < 1$, the series converges absolutely.
- If $L > 1$, the series diverges.
- If $L = 1$, the test is inconclusive.
Also, keep in mind the standard form of a geometric series for finding power series representations:
$$ \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}, \quad \text{for } |r| < 1 $$
Download Materials
Below you will find the blank practice test and the full worked-out solutions. I highly recommend attempting the problems on your own before peeking at the answer key!
Note: You may skip question #14 for this assignment.
Good luck studying! Recognizing which convergence test to use is half the battle—practice is the only way to build that intuition.