Practice Exam 3 – Infinite Series and Convergence Tests (Calculus II)

Q1(a). Evaluate the infinite series or state that the series diverges:

$\displaystyle \sum_{k=1}^{\infty} \frac{3k - 5k}{10k}$

Background

Topic: Infinite Series, Series Convergence

This question is testing your ability to analyze and evaluate an infinite series, or determine if it diverges.

Key Terms and Formulas:

• Infinite Series: $\displaystyle \sum_{k=1}^{\infty} a_k$

• Convergence: A series converges if the sequence of partial sums approaches a finite limit.

• Divergence: A series diverges if the sequence of partial sums does not approach a finite limit.

Step-by-Step Guidance

1. First, simplify the numerator $3k - 5k$ to combine like terms.

2. Write the simplified form of the general term $a_k$.

3. Check if the resulting series is a geometric series, a $p$-series, or another recognizable type.

4. Apply the appropriate convergence test based on the form of the series.

Try solving on your own before revealing the answer!

Q1(b). Evaluate the infinite series or state that the series diverges:

$\displaystyle \sum_{k=1}^{\infty} 3(1.001)^k$

Background

Topic: Geometric Series

This question is testing your ability to recognize and analyze a geometric series and determine its convergence or divergence.

Key Terms and Formulas:

• Geometric Series: $\displaystyle \sum_{k=1}^{\infty} ar^k$

• Convergence: A geometric series converges if $|r| < 1$.

• Sum Formula (if convergent): $S = \frac{a}{1 - r}$

Step-by-Step Guidance

1. Identify the first term $a$ and the common ratio $r$ in the series.

2. Check the value of $|r|$ to determine if the series converges or diverges.

3. If the series converges, set up the sum formula for a geometric series.

4. If the series diverges, state the reason based on the value of $r$.

Try solving on your own before revealing the answer!

Q2. Express $0.314141414\ldots$ as a ratio of two integers.

Background

Topic: Converting Repeating Decimals to Fractions

This question is testing your ability to express a repeating decimal as a rational number (fraction).

Key Terms and Formulas:

• Repeating Decimal: A decimal in which a digit or group of digits repeats infinitely.

• Let $x$ represent the repeating decimal, then use algebraic manipulation to solve for $x$ as a fraction.

Step-by-Step Guidance

1. Let $x = 0.314141414\ldots$

2. Identify the repeating part and determine how many digits repeat.

3. Multiply $x$ by a power of 10 so that the repeating part aligns after the decimal point.

4. Set up an equation by subtracting the original $x$ from the shifted version to eliminate the repeating part.

Try solving on your own before revealing the answer!

Q3. Use the Divergence Test for the infinite series $\displaystyle \sum_{k=1}^{\infty} \frac{2k^4 + k^3 + 1}{3k^4 + 4}$

Background

Topic: Divergence Test (Test for Divergence)

This question is testing your ability to use the Divergence Test to determine if a series diverges.

Key Terms and Formulas:

• Divergence Test: If $\lim_{k \to \infty} a_k \neq 0$, then $\sum a_k$ diverges.

• If $\lim_{k \to \infty} a_k = 0$, the test is inconclusive.

Step-by-Step Guidance

1. Identify the general term $a_k = \frac{2k^4 + k^3 + 1}{3k^4 + 4}$.

2. Compute $\lim_{k \to \infty} a_k$ by dividing numerator and denominator by the highest power of $k$ present.

3. Simplify the limit expression as $k$ approaches infinity.

4. Use the result to determine if the series diverges or if the test is inconclusive.

Try solving on your own before revealing the answer!

Q4. Use the Limit Comparison Test and the comparison series $\displaystyle \sum_{k=1}^{\infty} \frac{1}{k^2}$ to determine whether $\displaystyle \sum_{k=1}^{\infty} \frac{2k^4 + k^3 + 1}{3k^6 + 4}$ converges.

Background

Topic: Limit Comparison Test

This question is testing your ability to use the Limit Comparison Test to determine the convergence of a series by comparing it to a known $p$-series.

Key Terms and Formulas:

• Limit Comparison Test: For $\sum a_k$ and $\sum b_k$ with $a_k, b_k > 0$, if $\lim_{k \to \infty} \frac{a_k}{b_k} = c$ where $0 < c < \infty$, then both series converge or both diverge.

• $p$-series: $\sum \frac{1}{k^p}$ converges if $p > 1$.

Step-by-Step Guidance

1. Identify $a_k = \frac{2k^4 + k^3 + 1}{3k^6 + 4}$ and $b_k = \frac{1}{k^2}$.

2. Set up the limit $\lim_{k \to \infty} \frac{a_k}{b_k}$.

3. Simplify the expression inside the limit.

4. Evaluate the limit to determine if it is a positive finite number.

Try solving on your own before revealing the answer!

Q5. Use the Comparison Test and the comparison series $\displaystyle \sum_{k=1}^{\infty} \frac{8}{k^2}$ to determine whether $\displaystyle \sum_{k=1}^{\infty} \frac{7 + \sin k}{k^2}$ converges.

Background

Topic: Direct Comparison Test

This question is testing your ability to use the Comparison Test to determine the convergence of a series by comparing it to a known convergent series.

Key Terms and Formulas:

• Direct Comparison Test: If $0 \leq a_k \leq b_k$ and $\sum b_k$ converges, then $\sum a_k$ converges.

• $p$-series: $\sum \frac{1}{k^p}$ converges if $p > 1$.

Step-by-Step Guidance

1. Identify $a_k = \frac{7 + \sin k}{k^2}$ and $b_k = \frac{8}{k^2}$.

2. Find the range of $7 + \sin k$ for all $k$.

3. Show that $a_k \leq b_k$ for all $k$.

4. State the convergence of the comparison series $\sum b_k$.

Try solving on your own before revealing the answer!

Q6. Use the Root Test to determine whether the infinite series $\displaystyle \sum_{k=1}^{\infty} \left( \frac{k}{2k + 3} \right)^k$ converges.

Background

Topic: Root Test

This question is testing your ability to apply the Root Test to determine the convergence of a series.

Key Terms and Formulas:

• Root Test: For $\sum a_k$, compute $L = \lim_{k \to \infty} \sqrt[k]{|a_k|}$.

• If $L < 1$, the series converges; if $L > 1$, the series diverges; if $L = 1$, the test is inconclusive.

Step-by-Step Guidance

1. Identify $a_k = \left( \frac{k}{2k + 3} \right)^k$.

2. Set up the Root Test limit: $L = \lim_{k \to \infty} \sqrt[k]{\left| \frac{k}{2k + 3} \right|^k}$.

3. Simplify the expression inside the limit.

4. Evaluate the limit $L$ to determine convergence or divergence.

Try solving on your own before revealing the answer!

Q7. Use the Ratio Test to determine whether the infinite series $\displaystyle \sum_{k=1}^{\infty} k e^{-k}$ converges.

Background

Topic: Ratio Test

This question is testing your ability to apply the Ratio Test to determine the convergence of a series.

Key Terms and Formulas:

• Ratio Test: For $\sum a_k$, compute $L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|$.

• If $L < 1$, the series converges; if $L > 1$, the series diverges; if $L = 1$, the test is inconclusive.

Step-by-Step Guidance

1. Identify $a_k = k e^{-k}$.

2. Compute $a_{k+1} = (k+1) e^{-(k+1)}$.

3. Set up the ratio $\left| \frac{a_{k+1}}{a_k} \right|$ and simplify.

4. Take the limit as $k \to \infty$ to find $L$.

Try solving on your own before revealing the answer!

Q8. Use the Integral Test to determine whether the infinite series $\displaystyle \sum_{k=1}^{\infty} \frac{e^k}{1 + e^{2k}}$ converges.

Background

Topic: Integral Test

This question is testing your ability to use the Integral Test to determine the convergence of a series by evaluating an improper integral.

Key Terms and Formulas:

• Integral Test: If $f(k) = a_k$ is positive, continuous, and decreasing for $k \geq N$, then $\sum a_k$ converges if and only if $\int_N^{\infty} f(x) dx$ converges.

Step-by-Step Guidance

1. Let $f(x) = \frac{e^x}{1 + e^{2x}}$ and check that $f(x)$ is positive, continuous, and decreasing for $x \geq 1$.

2. Set up the improper integral $\int_1^{\infty} \frac{e^x}{1 + e^{2x}} dx$.

3. Consider substitution to simplify the integral (e.g., $u = e^x$).

4. Analyze the convergence of the resulting integral.

Try solving on your own before revealing the answer!

Q9(a). Determine whether the series $\displaystyle \sum_{k=1}^{\infty} \frac{(-1)^k}{k^{5/2}}$ converges absolutely, converges conditionally, or diverges.

Background

Topic: Absolute and Conditional Convergence of Alternating Series

This question is testing your ability to determine the type of convergence for an alternating series.

Key Terms and Formulas:

• Absolute Convergence: $\sum |a_k|$ converges.

• Conditional Convergence: $\sum a_k$ converges, but $\sum |a_k|$ diverges.

• Alternating Series Test: If $a_k$ decreases to 0, the series converges.

• $p$-series: $\sum \frac{1}{k^p}$ converges if $p > 1$.

Step-by-Step Guidance

1. Identify the general term $a_k = \frac{(-1)^k}{k^{5/2}}$.

2. Consider the absolute value $|a_k| = \frac{1}{k^{5/2}}$ and determine if this $p$-series converges.

3. If the absolute series converges, the original series converges absolutely.

4. If not, check for conditional convergence using the Alternating Series Test.

Try solving on your own before revealing the answer!

Q9(b). Determine whether the series $\displaystyle \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{3/7}}$ converges absolutely, converges conditionally, or diverges.

Background

Topic: Absolute and Conditional Convergence of Alternating Series

This question is testing your ability to determine the type of convergence for an alternating series.

Key Terms and Formulas:

• Absolute Convergence: $\sum |a_k|$ converges.

• Conditional Convergence: $\sum a_k$ converges, but $\sum |a_k|$ diverges.

• Alternating Series Test: If $a_k$ decreases to 0, the series converges.

• $p$-series: $\sum \frac{1}{k^p}$ converges if $p > 1$.

Step-by-Step Guidance

1. Identify the general term $a_k = \frac{(-1)^{k+1}}{k^{3/7}}$.

2. Consider the absolute value $|a_k| = \frac{1}{k^{3/7}}$ and determine if this $p$-series converges.

3. If the absolute series diverges, check for conditional convergence using the Alternating Series Test.

4. Analyze whether $a_k$ decreases to 0 and meets the conditions for the Alternating Series Test.

Try solving on your own before revealing the answer!