1. Definitions

1.1 Sequences

A sequence is a function mapping (the set of positive integers) to real numbers. A sequence converges to if for every , there exists such that for all .

• Notation:

• Divergence: If a sequence does not approach a finite limit, it diverges.

• Example: converges to $0n \to \infty$.

1.2 Series

A series is the sum of the terms of a sequence: . The series converges if the sequence of partial sums converges as .

• Convergence: converges if exists and is finite.

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

• Example: The geometric series converges if .

1.3 Improper Integrals

An improper integral is an integral where either the interval of integration is infinite or the integrand becomes infinite within the interval. To evaluate, we use limits.

• Type 1: Infinite interval:

• Type 2: Discontinuous integrand: where is not defined at some point in (e.g., $f(x)$ has a vertical asymptote).

• Convergence: The improper integral converges if the corresponding limit exists and is finite.

• Example: converges if and diverges if .

2. Integration

Integration is the process of finding the area under a curve, represented by the definite or indefinite integral. Key techniques include substitution, integration by parts, partial fractions, and recognizing standard forms.

• Definite Integral: gives the net area between and the -axis from to .

• Indefinite Integral: represents the family of all antiderivatives of .

• Improper Integrals: See Section 1.3 above.

• Example:

Practice Problems

3. Sequences

Sequences are ordered lists of numbers, often defined by a formula. Determining the limit of a sequence is a fundamental skill in calculus.

• Limit of a Sequence:

• Convergence Tests: Use algebraic manipulation, squeeze theorem, or known limits.

• Example: converges to $0$.

Practice Problems

4. Series

Series are sums of sequences. Determining convergence or divergence is essential, using various tests such as the comparison test, ratio test, root test, and alternating series test.

• Geometric Series: converges if .

• p-Series: converges if .

• Alternating Series Test: If decreases to $0\sum (-1)^n a_n$ converges.

• Ratio Test: . If , the series converges; if , diverges; if , inconclusive.

• Root Test: . Same conclusions as ratio test.

Practice Problems

5. Power Series

A power series is a series of the form , where is the center. The radius of convergence is the distance from within which the series converges.

• Finding Radius of Convergence: Use the ratio or root test to find .

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

• Example: converges for .

Practice Problems

1. Find the radius of convergence for

2. Find the radius of convergence for

3. Find the radius of convergence for

4. Find the radius of convergence for

5. Find the radius of convergence for

Additional Problems

1. Find the power series of and determine its radius of convergence.

2. Find the power series of and determine its radius of convergence.

Remember to study broadly and review all material covered in this course!