Let's tackle each question in turn.

Question 1

We need to show that the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges.

Solution:

This series is a p-series, which is given by $\sum_{n=1}^{\infty} \frac{1}{n^p}$, where $p$ is a positive constant. The convergence of a p-series depends on the value of $p$:

• If $p > 1$, the series converges.
• If $0 < p \leq 1$, the series diverges.

In our case: $p = 2 > 1$ Therefore, $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges.

To formally confirm this, we can compare it to the integral test for convergence. The integral test states that for a positive, decreasing, continuous function $f(x) = \frac{1}{x^2}$, the convergence of the integral $\int_1^{\infty} \frac{1}{x^2} \, dx$ implies the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$.

Calculating the integral: $\int_1^{\infty} \frac{1}{x^2} \, dx = \left[ -\frac{1}{x} \right]_1^{\infty} = 1$ Since the integral converges, by the integral test, the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges as well.

Question 2

We need to show that if $\sum_{n=1}^{\infty} |a_n| < \infty$, then $\sum_{n=0}^{\infty} a_n$ is convergent.

Solution:

This question is about the absolute convergence of a series.

1. Absolute Convergence Implies Convergence: If $\sum_{n=1}^{\infty} |a_n|$ converges, we say that the series $\sum_{n=1}^{\infty} a_n$ is absolutely convergent.
2. A well-known result in real analysis states that if a series is absolutely convergent, then it is also convergent.

Therefore, if $\sum_{n=1}^{\infty} |a_n|$ converges, it implies that $\sum_{n=1}^{\infty} a_n$ also converges.

Summary

1. For Question 1, we showed that $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges because it is a p-series with $p = 2 > 1$.
2. For Question 2, we applied the fact that absolute convergence implies convergence, so if $\sum_{n=1}^{\infty} |a_n|$ converges, then $\sum_{n=0}^{\infty} a_n$ also converges.

Would you like further details on any of these steps, or do you have additional questions?

Related Questions

1. What is the difference between conditional and absolute convergence?
2. How does the comparison test work in determining the convergence of a series?
3. Can the integral test be used to prove divergence as well as convergence?
4. What are some examples of p-series that diverge?
5. How does the ratio test apply to absolute convergence?

Tip

When analyzing series, always check if it's a well-known type (like geometric or p-series) first, as this can often immediately reveal its convergence properties.