Calculus 3: Applications and Advanced Uses

Introduction

• Review of finding derivatives and integrals.
• Focus on applications in Chapters 5 & 6: Sequences and Series.
• Introduction to Sequences and Series.
• Emphasis on infinite aspects.

Sequences

Definition

• An ordered list of numbers (e.g., 1, 4, 9, 16).
• Labels typically: $a_1, a_2, a_3,...$
• Interested in infinite sequences (no last term).

Representation

1. Explicit Formula: Defines each term, e.g., $a_n = n^2$.
2. Recurrence Relation: Each term depends on the previous term, e.g., $a_1 = 1$, $a_n = a_{n-1} + 2n - 1$.

Main Types of Sequences

1. Arithmetic Sequence
   • Common difference added to each term.
   • Example: 5, 8, 11, 14 (common difference = 3).
   • Representations:
     • Recursion: $a_1 = 5$, $a_n = a_{n-1} + d$
     • Explicit: $a_n = d(n-1) + ext{first term}$.
2. Geometric Sequence
   • Common ratio multiplied to each term.
   • Example: 4, -12, 36 (common ratio = -3).
   • Representations:
     • Recursion: $a_1 = 4$, $a_n = r imes a_{n-1}$
     • Explicit: $a_n = (r)^{(n-1)} imes ext{first term}$

Identifying Patterns

Alternating Sequences

1. Example: -1/2, 2/3, -3/4, 4/5
   • Alternating signs: use $(-1)^n$.
   • Numerators: counting numbers.
   • Denominators: one more than term number.
   • Formula: $a_n = (-1)^n rac{n}{n+1}$

Complex Sequences

1. Example: 3/4, 9/7, 27/10
   • Numerators: powers of 3.
   • Denominators: Arithmetic sequence.
   • Formula: $a_n = rac{3^n}{3n+1}$
2. Example: 2/2, -4/10, 12/50
   • Alternating signs: adjust exponent by addition of 1.
   • Numerator: Complex factorials.
   • Denominator: Multiplying series.
   • Reduced Formula Example: $a_n = (-1)^{n+1} rac{n!}{5^{n-1}}$

Limits and Convergence

Definitions

• Limit of a sequence: Check if terms approach a finite value (L) as $n o ext{infinity}$.
• Convergent Sequence: If it approaches L.
• Divergent Sequence: If it does not approach L.

Examples

• Limit approaching 1: $a_n = 1 - (1/2)^n$.
• Limit diverging: $a_n = 1 + 3n$ (approaches infinity).
• Using large exponents: $rac{3n^4 - 7n^2 + 5}{6 - 4n^4} o -3/4$.
• L'Hôpital’s Rule help when limits lead to $rac{ ext{Infinity}}{ ext{Infinity}}$ situations.

Bound and Monotone Sequences

Bounded Sequences

• Upper bound: Terms never exceed a number.
• Lower bound: Terms never go below a number.
  • Example: $a_n = rac{1}{n}$ bounded between 0 and 1.

Monotone Convergence Theorem

• Increasing and Upper bound: Sequence converges.
• Decreasing and Lower bound: Sequence converges.

Series (Sum of Sequences)

Infinite Series

• Definition: Sum of a sequence over all its terms.
• Notation: $ ext{Sum} = ext{Sum from }n=1 ext{ to infinity of } a_n$.

Convergence/Divergence

• Series convergence if Partial sums ($s_k = ext{Sum from } n=1 ext{ to } k$) converge.
  • Example: Harmonic series $rac{1}{n}$ diverges but alternating harmonic converges.
  • Geometric series: sum depends on |r| < 1

Special Series Examples

1. Harmonic Series: $rac{1}{n}$.
   • Divergent even though slowly increasing.
2. Geometric Series: $a R^{(n-1)}$.
   • Converges if $|r| < 1$.
3. Telescoping Series: Terms subtract out.
   • Example: $ ext{Sum}( ext{Cos}(1/n) - ext{Cos}(1/(n+1)))$.

Tests for Divergence/Convergence

Divergence Test

• If limit of terms $a_n$ is not zero, series diverges.

Integral Test

• Integrate functions similar to series.
  • If integral diverges, series diverges too.
  • Only for positive terms.
  • Example: $1/n^4$ convergent as integral from 1 to infinity of $1/n^4$ is finite.

P-Series Test

• $rac{1}{n^p}$ diverges for $p ext{≤} 1$, converges for $p ext{> }1$.

Comparison Test

• Direct Comparison: Compare terms $a_n$ with a known convergent/divergent series $b_n$.
  • If $a_n ext{≤} b_n$ and $ ext{Sum } b_n$ converges, then $ ext{Sum } a_n$ converges.
• Limit Comparison: Limit of $rac{a_n}{b_n}$.
  • If limit equals a finite number other than zero or infinity, both series behave similarly.

Alternating Series Test

• Conditions:
  1. Terms decrease in absolute value.
  2. Limit of terms is zero.
• If true, series converges.

Absolute vs Conditional Convergence

• Absolute Convergence: If $ ext{Sum of } |a_n|$ converges, then $ ext{Sum of } a_n$ converges.
• Conditional Convergence: Converges only when alternating.

Ratio and Root Tests

Ratio Test

• $ ho$ = Limit of absolute value of $rac{a_{n+1}}{a_n}$.
  • Converges absolutely if $ ho ext{< }1$.
  • Diverges if $ ho ext{> }1$.
  • Inconclusive if $ ho ext {= } 1$.

Root Test

• $ ho$ = Limit of $n^{th}$ root of absolute value of $a_n$.
  • Converges if $ ho ext{< }1$.
  • Diverges if $ ho ext{> }1$.
  • Inconclusive if $ ho ext {= } 1$.

Power Series & Use in Calculus

Definition

• Power series centered at x = 0: $ ext{Sum from} n=0 ext{ to infinity} of c_n x^n$.
• Centered at x = a: $ ext{Sum from} n=0 ext{ to infinity of } c_n (x-a)^n$.

Convergence

• Converge at x = a.
• Converge for all x: $ ext{Sum from} negative ext{infinity to} ext{positive infinity.}$
• Converge within a radius: Such that $|x - a| < R$

Finding Equations

1. Power Series of $f(x) = 1/(1 +x^3)$.
2. Converting feature equations into Power Series.
3. GEOMETRIC SERIES: Represents complex functions.
   • Example: $f(x) = x^2 / (4-x^2)$.

Working with and Converting Power Series

Derivatives/Integrals

• Derivative of each term: Easy derivative applications.
• Integrate each term: Include constant terms.
• Conversion to known forms useful, e.g., using Taylor series for e^x, sin(x).

Other Series

Taylor & Maclaurin Series

• Taylor Series: Series at $x=a$: $ ext{Sum from n=0 to infinity of } f^{(n)}(a)/n!*(x-a)^n$.
• Maclaurin Series: Taylor series centered at $a=0$.
  • Example: $e^x, ext{sin}(x)$.

Strategy

• Build patterns and functions through derivatives.
• Derivatives set initial conditions and solve for coefficients.

Converting series formulas

• Using finite series equivalency. \n#### Conclusion
• Extensive use of sequences and series for calculus applications.
• Tests and conversions necessary to gauge convergence.
• Power series beneficial to represent complex functions.