Sequences & series of functions

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   Consider the series $\sum_{n=1}^{\infty} \dfrac{2}{3^{n}}\sin(nx)$ on $\mathbb{R}$. The Weierstrass $M^-$test applies with $M_n = \dfrac{2}{3^{n}}$, since $|\sin(nx)| \le 1$ for every $x$. Compute $\sum_{n=1}^{\infty} M_n$ (the $M^-$test bound certifying uniform convergence).

2. Warm-up

   Consider the series $\sum_{n=1}^{\infty} \dfrac{3}{5^{n}}\sin(nx)$ on $\mathbb{R}$. The Weierstrass $M^-$test applies with $M_n = \dfrac{3}{5^{n}}$, since $|\sin(nx)| \le 1$ for every $x$. Compute $\sum_{n=1}^{\infty} M_n$ (the $M^-$test bound certifying uniform convergence).

3. Warm-up

   Consider the series $\sum_{n=1}^{\infty} \dfrac{6}{2^{n}}\sin(nx)$ on $\mathbb{R}$. The Weierstrass $M^-$test applies with $M_n = \dfrac{6}{2^{n}}$, since $|\sin(nx)| \le 1$ for every $x$. Compute $\sum_{n=1}^{\infty} M_n$ (the $M^-$test bound certifying uniform convergence).

Problems shown under their original terms: Generated (template, solver-computed).