SUMMARY
The series \(\sum_{n=1}^\infty \sin\left(\frac{1}{n^2}\right)\) converges. This conclusion is reached by applying the comparison test, specifically comparing it to the convergent series \(\sum_{n=1}^\infty \frac{1}{n^2}\). Since \(|\sin(x)| \leq x\) for \(x \geq 0\), it follows that \(\sin\left(\frac{1}{n^2}\right) \leq \frac{1}{n^2}\). Thus, the series converges due to the properties of nonnegative terms in convergence tests.
PREREQUISITES
- Understanding of series convergence tests, particularly the comparison test.
- Familiarity with the properties of the sine function, specifically \(|\sin(x)| \leq x\).
- Knowledge of the p-series test, particularly the convergence of \(\sum_{n=1}^\infty \frac{1}{n^2}\).
- Basic calculus concepts, including limits and inequalities.
NEXT STEPS
- Study the comparison test in detail, including its applications and limitations.
- Learn about the limit comparison test and how it can be applied to series.
- Explore the convergence of other trigonometric series for broader understanding.
- Investigate the behavior of \(\sin(x)\) for small values of \(x\) to understand its implications in series.
USEFUL FOR
Students studying calculus, particularly those focusing on series and convergence tests, as well as educators looking for examples of series convergence using trigonometric functions.