To prove that the given function

$f(x) = \sum_{n=0}^\infty a_n x^n$

is continuous on $[-2, 2]$, under the condition that $\sum_{n=0}^\infty 2^n |a_n|$ converges absolutely, let's proceed with the following steps:

1. Absolute Convergence Implies Uniform Convergence

The absolute convergence of $\sum_{n=0}^\infty 2^n |a_n|$ implies that the series of terms $a_n x^n$ is absolutely convergent for $|x| \leq 2$. To show continuity, we use the fact that a power series converges uniformly on closed intervals within its radius of convergence.

Radius of Convergence:

Using the ratio test: $\text{If } \sum_{n=0}^\infty 2^n |a_n| \text{ converges, then } \limsup_{n \to \infty} \sqrt[n]{2^n |a_n|} < \infty.$ This implies that the radius of convergence of $\sum_{n=0}^\infty a_n x^n$ is at least 2, ensuring the series converges absolutely for all $x \in [-2, 2]$.

Uniform Convergence:

On any closed subinterval of $[-2, 2]$, the Weierstrass M-test ensures uniform convergence because the terms $|a_n x^n| \leq M_n |x|^n$ (where $|a_n| \cdot 2^n$ is bounded).

2. Uniform Convergence Preserves Continuity

Since the series converges uniformly on $[-2, 2]$, it follows that $f(x)$ is continuous on $[-2, 2]$. This is a standard result for power series: uniform convergence implies that the sum of the series is continuous on the interval of uniform convergence.

3. Conclusion

The function $f(x)$ is continuous on $[-2, 2]$ because:

• The series $\sum_{n=0}^\infty a_n x^n$ converges absolutely for all $x \in [-2, 2]$.
• Absolute convergence ensures uniform convergence on compact subsets, such as $[-2, 2]$.
• Uniform convergence of the series implies the continuity of the function.

Let me know if you'd like further clarification on any of the steps!

Here are five related questions to expand your understanding:

1. What is the relationship between absolute convergence and uniform convergence?
2. How does the Weierstrass M-test help in proving uniform convergence?
3. What is the importance of the radius of convergence for a power series?
4. Can a power series be discontinuous within its interval of convergence? Why or why not?
5. What happens if the condition $\sum_{n=0}^\infty 2^n |a_n|$ does not hold?

Tip: For any power series, verifying the radius of convergence is a key step to determine where the series converges and behaves well!