Let’s address these problems step-by-step.

Problem 1: Convergence of the Power Series

$\sum_{n=1}^\infty (-1)^n \frac{4^{2n}(x-2)^n}{\sqrt{n}}$

Solution:

1. Determine absolute convergence:

   • Analyze $\left|a_n\right| = \frac{4^{2n}|x-2|^n}{\sqrt{n}}$.
   • To check convergence, use the Ratio Test: $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{4^{2(n+1)}|x-2|^{n+1}/\sqrt{n+1}}{4^{2n}|x-2|^n/\sqrt{n}} = \lim_{n \to \infty} \frac{16|x-2|}{\sqrt{n+1}/\sqrt{n}}$
     • As $n \to \infty$, the ratio simplifies to $16|x-2|$.
     • For convergence, $16|x-2| < 1$. Thus, $|x-2| < \frac{1}{16}$.

2. Interval of convergence:

   • Test endpoints:
     • At $x = 2 + \frac{1}{16}$ or $x = 2 - \frac{1}{16}$, the series becomes an alternating series. Analyze $\sqrt{n}$ growth to check conditional convergence.

3. Conclusion:

   • The series converges absolutely for $|x-2| < \frac{1}{16}$ and conditionally at the boundaries if alternating series criteria are satisfied.

Problem 2: Sum of the Infinite Series

$3 + \frac{9}{2!} + \frac{27}{3!} + \frac{81}{4!} + \cdots$

Solution:

1. Recognize the pattern:
   The series can be written as: $\sum_{n=0}^\infty \frac{3^{n+1}}{n!}$ Factor out the 3: $3 \sum_{n=0}^\infty \frac{3^n}{n!}$

2. Use the exponential series formula: The series $\sum_{n=0}^\infty \frac{x^n}{n!} = e^x$. Thus: $\sum_{n=0}^\infty \frac{3^n}{n!} = e^3$

3. Final result: $3 \cdot e^3$

Problem 3: Power Series to Find Limit

$\lim_{x \to 0} \frac{e^{-x^2} - 1 + x^2}{5x^4}$

Solution:

1. Expand $e^{-x^2}$ as a power series: $e^{-x^2} = 1 - x^2 + \frac{x^4}{2!} - \frac{x^6}{3!} + \cdots$

2. Subtract and simplify: $e^{-x^2} - 1 + x^2 = \frac{x^4}{2!} - \frac{x^6}{3!} + \cdots$

3. Divide by $5x^4$: $\frac{\frac{x^4}{2!} - \frac{x^6}{3!} + \cdots}{5x^4} = \frac{1}{10} - \frac{x^2}{30} + \cdots$

4. Take the limit as $x \to 0$: $\lim_{x \to 0} \frac{e^{-x^2} - 1 + x^2}{5x^4} = \frac{1}{10}$

Problem 4: Indefinite Integral as Power Series

$\int \frac{\sin(x^2)}{x} dx$

Solution:

1. Expand $\sin(x^2)$ as a power series: $\sin(x^2) = x^2 - \frac{x^6}{3!} + \frac{x^{10}}{5!} - \cdots$

2. Divide by $x$: $\frac{\sin(x^2)}{x} = x - \frac{x^5}{3!} + \frac{x^9}{5!} - \cdots$

3. Integrate term by term: $\int \frac{\sin(x^2)}{x} dx = \frac{x^2}{2} - \frac{x^6}{6 \cdot 3!} + \frac{x^{10}}{10 \cdot 5!} - \cdots$

4. Estimate the integral $\int_0^{0.1} \frac{\sin(x^2)}{x} dx$ with error no greater than $10^{-9}$:

   • Calculate sufficient terms until the error is $\leq 10^{-9}$.

Would you like detailed clarification on any of these steps?

Follow-up Questions:

1. How does the Ratio Test help in determining absolute convergence in Problem 1?
2. What is the role of exponential series in solving Problem 2?
3. Why is the power series expansion crucial in solving limits as $x \to 0$ in Problem 3?
4. How does term-by-term integration apply to Problem 4?
5. What methods ensure error bounds in numerical series approximations?

Tip: Always verify convergence before approximating infinite series or integrals to avoid misleading results.