etermine whether the following series converge or diverge. If they converge, you do not need to find
the sum. Justify your answers by stating which test(s) you’re using, or any relevant facts about conveg-
ent/divergent series (eg. geometric/p-series, etc.).
(a)
∞X
k=2
1
k√2 − 1
(b)
∞X
k=1
r k
k3 + 1
(c)
∞X
k=0
2k+2 + k2
3k + k + 12
(d)
∞X
k=3
1
k2 ln(k)

To determine whether each series converges or diverges, we will apply appropriate convergence tests. The most common tools include the **comparison test**, **limit comparison test**, **ratio test**, **root test**, and knowledge of well-known convergent/divergent series like the **p-series** or **geometric series**. Here’s the analysis for each:

---

### (a) \( \sum_{k=2}^{\infty} \frac{1}{k^{\sqrt{2}} - 1} \)

#### Test Used: Comparison Test

Let's compare this series to a simpler series. As \( k \to \infty \), the term \( k^{\sqrt{2}} - 1 \) behaves very similarly to \( k^{\sqrt{2}} \), since the "-1" becomes insignificant for large \( k \). Thus, for large \( k \), the term \( \frac{1}{k^{\sqrt{2}} - 1} \) behaves like \( \frac{1}{k^{\sqrt{2}}} \).

Now consider the series \( \sum \frac{1}{k^{\sqrt{2}}} \), which is a p-series with \( p = \sqrt{2} \). Since \( \sqrt{2} > 1 \), this p-series converges.

By comparison, \( \sum \frac{1}{k^{\sqrt{2}} - 1} \) converges as well.

#### Conclusion: **Converges**

---

### (b) \( \sum_{k=1}^{\infty} \frac{r^k}{k^3 + 1} \) (where \( 0 \leq r < 1 \))

#### Test Used: Ratio Test (and knowledge of geometric series)

This series contains a geometric component \( r^k \) and a decaying term \( \frac{1}{k^3 + 1} \).

1. For large \( k \), \( k^3 + 1 \sim k^3 \), so \( \frac{1}{k^3 + 1} \) behaves like \( \frac{1}{k^3} \).
2. Consider just the geometric component: \( r^k \) decays exponentially since \( 0 \leq r < 1 \).
3. The \( k^3 \) term in the denominator decays polynomially.

Since \( r^k \) decays exponentially, this decay dominates the polynomial growth of \( k^3 + 1 \), and the series converges by the **ratio test**.

#### Conclusion: **Converges**

---

### (c) \( \sum_{k=0}^{\infty} \frac{2^{k+2} + k^2}{3^k + k + 12} \)

#### Test Used: Dominant Term Analysis (Asymptotic Behavior)

For large \( k \), the dominant terms in the numerator and denominator are:
- Numerator: \( 2^{k+2} \sim 2^k \)
- Denominator: \( 3^k \)

Thus, for large \( k \), the terms in the series behave like \( \frac{2^k}{3^k} \).

Simplifying this expression:
\[ \frac{2^k}{3^k} = \left(\frac{2}{3}\right)^k \]

This is a geometric series with a ratio \( \frac{2}{3} \), which is less than 1. A geometric series with \( |r| < 1 \) converges.

#### Conclusion: **Converges**

---

### (d) \( \sum_{k=3}^{\infty} \frac{1}{k^2 \ln(k)} \)

#### Test Used: Integral Test

We use the **integral test** here, since this series involves a function that is continuous, positive, and decreasing for large \( k \).

The corresponding integral to evaluate is:
\[ \int_{3}^{\infty} \frac{1}{x^2 \ln(x)} dx \]

Using integration by substitution (with \( u = \ln(x) \), so \( du = \frac{1}{x} dx \)), we find that the integral converges. Therefore, by the **integral test**, the series also converges.

#### Conclusion: **Converges**

---

### Summary:

- (a) **Converges**
- (b) **Converges**
- (c) **Converges**
- (d) **Converges**

---

Would you like any details or further explanations on any of the tests used? Here are five related questions:

1. How does the comparison test work, and when is it applicable?
2. What is the p-series test, and how do we determine convergence for different values of \( p \)?
3. Can the ratio test be used for series with more complicated terms? How do we apply it?
4. What is the integral test, and why does it work for some series but not others?
5. How can asymptotic analysis help in determining the behavior of terms in a series?

**Tip**: When using the ratio test, always check the limit of the ratio as \( k \to \infty \), and remember that the test works well when the series behaves like a geometric series.

Determine whether the following series converge or diverge. If they converge, you do not need to find the sum. Justify your answers by stating which test(s) you're using, or any relevant facts about convergent/divergent series (e.g., geometric/p-series, etc.). (a) ∞Σk=2 1/(k^√2 - 1), (b) ∞Σk=1 r^k/(k^3 + 1), (c) ∞Σk=0 (2^(k+2) + k^2)/(3^k + k + 12), (d) ∞Σk=3 1/(k^2 ln(k))

Explore whether the series Σ(1/(k^√2 - 1)), Σ(r^k/(k^3 + 1)), Σ((2^(k+2) + k^2)/(3^k + k + 12)), and Σ(1/(k^2 ln(k))) converge or diverge. Learn about key convergence tests, such as the comparison test, integral test, and geometric series analysis. Ideal for college-level students studying advanced calculus.

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