Calculus II Series and Sequences Review – Step-by-Step Study Guidance

Q1. Write the Maclaurin Series for the function $f(x) = x \cos(x)$. Then find the interval and radius of convergence.

Background

Topic: Power Series, Maclaurin Series, Interval and Radius of Convergence

This question tests your ability to construct the Maclaurin series (a Taylor series centered at $x=0$) for a given function, and to determine where the series converges.

Key Terms and Formulas:

• Maclaurin Series: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$

• Radius of Convergence ($R$): The distance from the center ($x=0$) within which the series converges.

• Interval of Convergence: The set of $x$-values for which the series converges.

Step-by-Step Guidance

1. Recall the Maclaurin series for $\cos(x)$: $\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}$.

2. Multiply the series for $\cos(x)$ by $x$ to get the series for $x \cos(x)$: $x \cos(x) = x \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}$.

3. Rewrite the product as a single power series: $x \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n+1}$.

4. To find the radius and interval of convergence, use the Ratio Test on the general term $a_n = \frac{(-1)^n}{(2n)!} x^{2n+1}$.

5. Set up the Ratio Test: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$ and simplify the expression to solve for $|x|$.

Try solving on your own before revealing the answer!

Q2. You are given the sequence $a_n = \frac{n}{n+1}$. Find the general term and determine whether it converges or diverges.

Background

Topic: Sequences, Limits, Convergence/Divergence

This question asks you to analyze a sequence, write its general term, and determine its behavior as $n \to \infty$.

Key Terms and Formulas:

• Sequence: An ordered list of numbers, often defined by a formula $a_n$.

• Convergence: A sequence converges if $\lim_{n \to \infty} a_n$ exists and is finite.

• Divergence: A sequence diverges if the limit does not exist or is infinite.

Step-by-Step Guidance

1. Write the general term: $a_n = \frac{n}{n+1}$.

2. To determine convergence, compute $\lim_{n \to \infty} \frac{n}{n+1}$.

3. Simplify the expression by dividing numerator and denominator by $n$.

Try solving on your own before revealing the answer!

Q3. For each of the following series, determine if it converges or diverges. If geometric or telescoping, find the sum. Use as many tests as possible.

Background

Topic: Series Convergence Tests (Geometric, Telescoping, Comparison, Ratio, etc.)

This question asks you to analyze several series for convergence/divergence and, if possible, find their sums.

Key Terms and Formulas:

• Geometric Series: $\sum_{n=0}^{\infty} ar^n$ converges if $|r| < 1$.

• Telescoping Series: Series where terms cancel in partial sums.

• Common Tests: Ratio Test, Root Test, Comparison Test, Alternating Series Test, etc.

Step-by-Step Guidance

1. For each series, identify its type (geometric, telescoping, $p$-series, etc.).

2. If geometric, find the common ratio $r$ and check if $|r| < 1$.

3. If telescoping, write out the first few terms to see cancellation.

4. If not geometric or telescoping, choose an appropriate test (Ratio, Root, Comparison, etc.) and set up the test.

5. For geometric or telescoping series, set up the formula for the sum if applicable.

Try solving on your own before revealing the answer!

Q4. Suppose you want to find the value of $\sin(5 \theta)$ up to the fourth term of its Maclaurin series. What would you do?

Background

Topic: Maclaurin Series, Approximations

This question asks you to use the Maclaurin series for $\sin(x)$ to approximate $\sin(5\theta)$ up to the fourth term.

Key Terms and Formulas:

• Maclaurin Series for $\sin(x)$: $\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1}$

Step-by-Step Guidance

1. Write out the first four nonzero terms of the Maclaurin series for $\sin(x)$.

2. Substitute $x = 5\theta$ into the series.

3. Write out the resulting expression up to the fourth term.

Try solving on your own before revealing the answer!

Q5. Write the Maclaurin polynomial to the fifth term for the function $f(x) = x \cos(a)$.

Background

Topic: Maclaurin Polynomials

This question asks you to write the Maclaurin polynomial (up to the fifth term) for a function involving a constant $a$.

Key Terms and Formulas:

• Maclaurin Polynomial: The sum of the first $n$ terms of the Maclaurin series.

Step-by-Step Guidance

1. Recall that $\cos(a)$ is a constant with respect to $x$.

2. Write the Maclaurin series for $f(x) = x \cos(a)$ by expanding $x$ as a power series (which is just $x$) and multiplying by $\cos(a)$.

3. List the first five terms of the resulting polynomial.

Try solving on your own before revealing the answer!

Q6. Use the ratio, difference, and derivative tests for sequences to determine if the sequence is strictly increasing, strictly decreasing, or neither.

Background

Topic: Monotonicity of Sequences

This question asks you to use various tests to determine if a sequence is increasing, decreasing, or neither.

Key Terms and Formulas:

• Ratio Test for Sequences: Compare $a_{n+1}$ and $a_n$.

• Difference Test: Analyze $a_{n+1} - a_n$.

• Derivative Test: If $a_n = f(n)$, check $f'(x)$ for monotonicity.

Step-by-Step Guidance

1. Write the general term of the sequence.

2. Compute $a_{n+1} - a_n$ and analyze its sign.

3. Alternatively, compute $\frac{a_{n+1}}{a_n}$ and check if it is greater or less than 1.

4. If the sequence is defined by a function $f(n)$, compute $f'(x)$ and analyze its sign for $x \geq 1$.

Try solving on your own before revealing the answer!

Q7. Prove that $e^{ix} = \cos(x) + i \sin(x)$ using the Maclaurin series for $e^x$.

Background

Topic: Euler's Formula, Maclaurin Series

This question asks you to use the Maclaurin series for $e^x$ to derive Euler's formula.

Key Terms and Formulas:

• Maclaurin Series for $e^x$: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$

• Euler's Formula: $e^{ix} = \cos(x) + i \sin(x)$

Step-by-Step Guidance

1. Write the Maclaurin series for $e^{ix}$ by substituting $ix$ for $x$ in the series for $e^x$.

2. Expand the powers of $i$ and group real and imaginary terms.

3. Recognize the real part as the Maclaurin series for $\cos(x)$ and the imaginary part as the series for $\sin(x)$.

Try solving on your own before revealing the answer!

Q8. Integrate the function $f(x) = x \cos(x)$ using its Maclaurin series.

Background

Topic: Series Integration

This question asks you to integrate a function by first expressing it as a Maclaurin series, then integrating term by term.

Key Terms and Formulas:

• Maclaurin Series for $x \cos(x)$: $x \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n+1}$

• Term-by-Term Integration: $\int \sum a_n x^n dx = \sum a_n \int x^n dx$

Step-by-Step Guidance

1. Write the Maclaurin series for $x \cos(x)$.

2. Integrate each term of the series with respect to $x$.

3. Write the resulting power series for the integral.

Try solving on your own before revealing the answer!

Q9. Integrate the function $f(x) = \ln|x|$ from $-1$ to $0$. If it does not work, please explain why.

Background

Topic: Improper Integrals, Logarithmic Functions

This question asks you to evaluate an integral with a possible discontinuity at the lower limit.

Key Terms and Formulas:

• Improper Integral: An integral where the function is unbounded or the interval is infinite.

• Logarithmic Properties: $\ln|x|$ is undefined at $x=0$.

Step-by-Step Guidance

1. Set up the definite integral $\int_{-1}^0 \ln|x| dx$.

2. Recognize that $\ln|x|$ is undefined at $x=0$, so the integral is improper at the upper limit.

3. Rewrite the integral as a limit: $\lim_{a \to 0^-} \int_{-1}^a \ln|x| dx$.

4. Attempt to evaluate the integral and analyze the behavior as $a \to 0^-$.

Try solving on your own before revealing the answer!

Q10. Write the Taylor series for $f(x) = \ln(x)$ at $x_0 = 1$.

Background

Topic: Taylor Series Expansion

This question asks you to write the Taylor series for $\ln(x)$ centered at $x=1$.

Key Terms and Formulas:

• Taylor Series at $x_0$: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(x_0)}{n!} (x - x_0)^n$

• Derivatives of $\ln(x)$: $f'(x) = 1/x$, $f''(x) = -1/x^2$, etc.

Step-by-Step Guidance

1. Compute the first few derivatives of $\ln(x)$ and evaluate them at $x=1$.

2. Write the general term for the Taylor series using these derivatives.

3. Write out the first few terms explicitly.

Try solving on your own before revealing the answer!