Calculus 2: Sequences, Series, and Integration by Parts

Calculus 2

Calculus 2 is a higher-level mathematics course that builds on the concepts learned in Calculus 1. In Calculus 2, students learn about advanced topics in calculus, including sequences and series, integration by parts, and more.

Sequences and Series

A sequence is a set of numbers listed in a specific order. Each number in a sequence is called a term. A series is the sum of the terms of a sequence.

Sequences

Sequences can be expressed in three different notations:

1. List notation: A sequence can be written as a list of numbers in parentheses, with each term separated by commas. For example, the sequence {1, 3, 5} has three terms.

2. List notation with arrows: A sequence can also be written as a list with arrows pointing to the right. For example, the sequence 1 → 3 → 5 has three terms.

3. Table notation: A sequence can be written in a table format, with each term listed in a separate row. For example:

   n	1	3	5
   1	1	3	5
   2	1	3	5
   3	1	3	5

   In this notation, the sequence is 1, 3, 5.

Series

A series is the sum of the terms of a sequence. For example, the series of the sequence {1, 3, 5} is 1 + 3 + 5 = 9.

Arithmetic Sequences

An arithmetic sequence is a sequence in which each term is obtained by adding a constant to the previous term. For example, the sequence 1, 3, 5 is an arithmetic sequence, because each term is obtained by adding 2 to the previous term.

Geometric Sequences

A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a constant. For example, the sequence 1, 2, 4 is a geometric sequence, because each term is obtained by multiplying the previous term by 2.

Convergence

A sequence is said to converge if it approaches a limit as n approaches infinity. In other words, if the terms in the sequence get closer and closer to a certain number as n increases, the sequence is said to converge to that number.

For example, the sequence {1, 1.5, 1.625, 1.6875, ...} converges to 2.

Integration by Parts

Integration by parts is a method for finding the indefinite integral of a product of two functions. It is based on the product rule for differentiation.

Product Rule

The product rule is a rule for differentiating a product of two functions. It states that:

If f(x) and g(x) are functions of x, then (f(x)g(x))' = f(x)g'(x) + g(x)f'(x).

Integration by Parts

Integration by parts can be thought of as the inverse of the product rule. It states that:

If f(x) and g(x) are functions of x, then ∫f(x)g(x) dx = (f(x)g(x)) + C, where C is the constant of integration.

Example

Consider the integral ∫x^2 sin x dx. Using integration by parts, we can write:

∫x^2 sin x dx = u du = x^2 (−cos x) + C

where u = x^2 and du = 2x dx.

In this example, we used the product rule to find that (x^2)(−cos x)' = x^2(−sin x) + 2x(cos x), and then used integration by parts to find that ∫x^2 sin x dx = ∫(x^2)(−sin x) dx = (x^2)(−cos x) + C.

Applications

Integration by parts finds many applications in physics and engineering. For example, it is used to find the work done by a variable force, to find the moment of inertia of a curved surface, and to solve the heat equation.

In conclusion, Calculus 2 is a course that delves into advanced topics such as sequences and series, integration by parts, and more. These topics are crucial for understanding advanced mathematics and can be applied to various fields, such as physics and engineering.