# Calculus II: Sequences, Series and Multivariable Analysis

> Course notes covering sequences, numerical series, multivariable functions, multiple integrals, line integrals and surface integrals.

- Author: Muhammet Ali Köker
- Language: en
- Canonical: https://alikoker.com.tr/en/calculus-ii-sequences-series-multivariable-analysis
- Translation: https://alikoker.com.tr/matematik-ii-diziler-seriler-cok-degiskenli-analiz
- Published: 2014-02-23T17:35:00+03:00
- Modified: 2026-08-09T00:12:00+03:00
- Verified: 2026-08-09T00:12:00+03:00
- Type: article

I prepared my Mathematics II notes during undergraduate Computer Engineering courses in the 2013-2014 period. This revision preserves the original course progression from sequences and series to multivariable functions, multiple integrals, line integrals, and surface integrals, while reviewing terminology and technical conditions with later sources.

## Unit 1: Limits of Sequences

### Definition of a numerical sequence

A numerical sequence is a function whose domain is usually the positive integers:

```text
a: N → R,   n ↦ a_n.
```

The index records order, so a sequence is not merely a set of values.

### Operations on sequences

Given sequences `(a_n)` and `(b_n)`, sums, differences, products, and quotients are defined term by term wherever denominators are nonzero. Scalar multiplication is also defined termwise.

### Bounded and unbounded sequences

A sequence is bounded above if there is an `M` with `a_n ≤ M` for every `n`, bounded below analogously, and bounded when both conditions hold. Boundedness alone does not imply convergence.

### Supremum and infimum of a sequence

The set of sequence values can have a least upper bound `sup a_n` and greatest lower bound `inf a_n`. These notions summarize bounds even when the extrema are not attained by a term.

### Limit of a sequence

The statement

```text
lim_(n→∞) a_n = L
```

means that for every `ε>0`, there is an index `N` such that

```text
|a_n - L| < ε
```

whenever `n≥N`.

### Properties of convergent sequences

A convergent sequence has a unique limit and is bounded. Standard algebraic limit laws apply to sums, products, and quotients under the usual denominator condition.

### Subsequences

A subsequence is obtained by choosing increasing indices `n_k`. Every subsequence of a convergent sequence converges to the same limit. Divergent sequences can nevertheless possess convergent subsequences.

### Monotone sequences

A monotone increasing sequence satisfies `a_(n+1) ≥ a_n`; a monotone decreasing sequence satisfies the reverse inequality. A bounded monotone sequence converges.

### Bolzano-Weierstrass theorem

Every bounded sequence in `R^n` has a convergent subsequence. In one dimension this connects boundedness with the existence of accumulation behavior even when the full sequence does not converge.

### Cauchy sequences

A sequence is Cauchy if its terms eventually become arbitrarily close to one another:

```text
for every ε>0, there is N such that
|a_n-a_m|<ε for n,m≥N.
```

In the real numbers, a sequence is convergent exactly when it is Cauchy. This equivalence relies on completeness.

## Unit 2: Numerical Series

### Definition of a series

A series

```text
Σ_(n=1)^∞ a_n
```

is studied through its sequence of partial sums

```text
S_N = Σ_(n=1)^N a_n.
```

The series converges if `(S_N)` converges.

### Convergent and divergent series

If the partial sums have a finite limit, the series is convergent; otherwise it diverges. Divergence can mean growth without bound, oscillation, or failure to approach one finite value.

### Necessary condition for convergence

A necessary condition is

```text
a_n → 0.
```

It is not sufficient. The harmonic series is the standard counterexample.

### Properties of convergent series

Finite changes to the initial terms do not affect convergence. Constant multiples and sums of convergent series remain convergent. Manipulating infinite series requires attention to convergence, especially when terms are rearranged.

### Remainder of a series

For a convergent series with sum `S`, the remainder after `N` terms is

```text
R_N = S - S_N.
```

Convergence tests and integral estimates can sometimes provide explicit bounds for `R_N`.

### Cauchy criterion for series

A series converges exactly when its tails can be made arbitrarily small:

```text
|a_(n+1)+...+a_m| < ε
```

for sufficiently large `m>n`.

### Positive-term series

When every term is nonnegative, the partial sums are monotone increasing. Such a series converges exactly when its partial sums are bounded above.

### Integral test

For a positive, continuous, decreasing function `f` with `a_n=f(n)`, the series `Σa_n` and the improper integral `∫f(x)dx` have the same convergence behavior. The integral can also bound the remainder.

### Comparison test

If `0≤a_n≤b_n` eventually and `Σb_n` converges, then `Σa_n` converges. If `a_n≥b_n≥0` eventually and `Σb_n` diverges, then `Σa_n` diverges.

### D'Alembert ratio test

For positive terms, if

```text
L = lim |a_(n+1)/a_n|,
```

then `L<1` implies absolute convergence and `L>1` implies divergence. The case `L=1` is inconclusive.

### Cauchy root test

If

```text
L = limsup |a_n|^(1/n),
```

then `L<1` gives absolute convergence and `L>1` divergence. Again, the boundary case requires another argument.

### Raabe test

Raabe's criterion refines the ratio test for some positive series when the ordinary ratio limit approaches one. Its usefulness lies in detecting more subtle decay rates.

### Alternating series

An alternating series of decreasing positive magnitudes tending to zero converges by the Leibniz criterion. The first omitted term bounds the magnitude of the remainder under these conditions.

### Absolute and conditional convergence

A series converges absolutely if

```text
Σ |a_n|
```

converges. Absolute convergence implies convergence. A series that converges but not absolutely is conditionally convergent.

### Riemann rearrangement theorem

A conditionally convergent real series can be rearranged to change its sum or even to diverge. This is why arbitrary rearrangement is safe for absolutely convergent series but not for conditionally convergent ones.

### Dirichlet and Abel tests

These tests establish convergence for products of sequences when one factor has controlled partial sums and the other has suitable monotone or bounded behavior. They are particularly useful for oscillatory series.

### Power series

A power series centered at `a` is

```text
Σ c_n (x-a)^n.
```

There is a radius `R` such that the series converges absolutely for `|x-a|<R` and diverges for `|x-a|>R`; endpoints must be tested separately.

### Taylor and Maclaurin series

For a sufficiently regular function, Taylor's series is built from derivatives at a point:

```text
Σ f^(n)(a)(x-a)^n/n!.
```

Maclaurin is the special case `a=0`. A formal Taylor series represents the function only where the remainder tends to zero.

### Fourier series

A Fourier series represents a periodic function through sine and cosine components. Coefficients are obtained from orthogonality integrals. Convergence at discontinuities requires care; pointwise behavior can differ from uniform behavior.

## Unit 3: Multivariable Functions

### Multivariable function and domain

A function such as

```text
f: R^2 → R,   z=f(x,y)
```

assigns one value to each point in a domain of the plane. Its graph is a surface in three-dimensional space.

### Limit and continuity

The limit

```text
lim_((x,y)→(a,b)) f(x,y)=L
```

must be the same along every path approaching `(a,b)`. Different path limits prove that a limit does not exist, although agreement along several paths alone does not prove existence.

Continuity requires the function value to exist and equal the limit.

### Partial derivatives

Partial differentiation changes one variable while holding the others fixed:

```text
f_x = ∂f/∂x,   f_y = ∂f/∂y.
```

Higher and mixed partial derivatives follow by repeated differentiation.

### Differentiability and total differential

Differentiability is stronger than the mere existence of partial derivatives. Near a differentiability point,

```text
Δf = f_x Δx + f_y Δy + o(||(Δx,Δy)||).
```

The linear part is the total differential.

### Chain rule

If `x` and `y` depend on `t`,

```text
df/dt = f_x dx/dt + f_y dy/dt.
```

The Jacobian generalizes this composition rule to vector-valued mappings.

### Homogeneous functions and Euler's theorem

If

```text
f(tx,ty)=t^k f(x,y),
```

then, under differentiability,

```text
x f_x + y f_y = k f.
```

### Implicit functions

An equation `F(x,y)=0` can locally define one variable as a function of another under an appropriate nonzero partial-derivative condition. Implicit differentiation then gives derivative information without solving explicitly.

### Jacobian

For a coordinate transformation `(u,v)→(x,y)`, the Jacobian determinant

```text
J = ∂(x,y)/∂(u,v)
```

measures local oriented area scaling. In multiple integrals its absolute value appears in change-of-variables formulas.

### Coordinate transformations

Cartesian, polar, cylindrical, and spherical coordinate systems simplify problems with matching geometry. A valid transformation must include the correct domain and Jacobian factor.

### Functional dependence

Functions can satisfy a relation that makes them not independent. Jacobian rank is one tool for studying local functional dependence.

### Taylor formula

A multivariable Taylor expansion uses partial derivatives to approximate a function near a point. The quadratic terms involve the Hessian matrix and are central to local extremum classification.

### Unconstrained extrema

At an interior differentiable local extremum,

```text
∇f = 0
```

is a necessary condition. The Hessian can classify nondegenerate critical points as minima, maxima, or saddles.

### Constrained extrema and Lagrange multipliers

For a constraint `g(x,y)=c`, a regular constrained extremum satisfies

```text
∇f = λ ∇g.
```

The method encodes the fact that allowable tangent directions cannot increase `f` to first order.

### Curves, tangent vector, and normal plane

For a parametric curve `r(t)`, the derivative `r'(t)` is a tangent vector when nonzero. A plane normal to the curve at a point has the tangent as its normal vector.

### Surfaces, tangent plane, and normal vector

For an implicit surface `F(x,y,z)=c`, the gradient

```text
∇F
```

is normal to the level surface where it is nonzero. The tangent plane is therefore defined from the gradient.

### Gradient

```text
∇f = (f_x,f_y,f_z)
```

points in the direction of steepest local increase, and its norm gives the maximum directional derivative.

### Directional derivative

For a unit vector `u`,

```text
D_u f = ∇f · u.
```

### Divergence, curl, and Laplacian

For a vector field `F`, divergence measures local source/sink behavior and curl measures local rotation. For a scalar field,

```text
Δf = ∇·∇f
```

is the Laplacian.

## Unit 4: Double Integrals

### Partition of a region and definition

A double integral extends Riemann integration to a planar region `D`:

```text
∬_D f(x,y) dA.
```

It is defined through limits of sums over small subregions when the required integrability conditions hold.

### Geometric meaning

For nonnegative `f`, the double integral can represent volume under the surface `z=f(x,y)` above the region `D`.

### Evaluation of double integrals

On suitable regions, Fubini's theorem allows the integral to be computed as iterated single integrals:

```text
∫ [∫ f(x,y) dy] dx
```

or in the reverse order.

### Properties

Linearity, additivity over subregions, comparison, and bounds extend naturally from one-dimensional integration.

### Transformations of regions

With a one-to-one differentiable coordinate transformation,

```text
dA = |J| du dv.
```

Polar coordinates give the familiar factor `r`.

### Mass, center of mass, and moment of inertia

If surface density is `ρ(x,y)`, then

```text
m = ∬_D ρ dA.
```

Weighted coordinate integrals give the center of mass; distance-squared weighted integrals give moments of inertia.

### Surface area

For a surface `z=f(x,y)` over `D`,

```text
A = ∬_D sqrt(1+f_x^2+f_y^2) dA.
```

### Volume of a solid of revolution

Double integrals can calculate volume by integrating a unit density over the appropriate planar or transformed region. The coordinate system should be selected from the geometry of the solid.

## Unit 5: Triple Integrals

### Definition and basic properties

A triple integral

```text
∭_V f(x,y,z) dV
```

extends integration to a three-dimensional region. Linearity and additivity hold as in lower dimensions.

### Evaluation

On suitable regions, the integral is evaluated as an iterated integral with bounds describing the solid. The order can be changed when the region and integrability conditions allow.

### Region transformations

Cylindrical and spherical coordinates are common transformations. The volume element must include the Jacobian:

```text
cylindrical: dV = r dr dθ dz
spherical:   dV = ρ^2 sinφ dρ dφ dθ.
```

### Volume, mass, center of mass, and moment of inertia

Taking `f=1` gives volume. Taking `f=ρ` gives mass. First moments determine center of mass; distance-squared moments determine moments of inertia.

## Unit 6: Line Integrals

### Line integral in the plane

For a curve `C` and scalar field `f`,

```text
∫_C f ds
```

accumulates the field along arc length. A parametrization `r(t)` converts it to an ordinary integral with the speed factor `||r'(t)||`.

### Line integrals with respect to coordinates

Expressions such as

```text
∫_C P dx + Q dy
```

are evaluated from a parametrization using `dx=x'(t)dt` and `dy=y'(t)dt`.

### Line integrals in space

The same construction extends to three-dimensional curves, where scalar or vector fields are integrated along `r(t)`.

### Line integral of a vector field

For a vector field `F`,

```text
∫_C F·dr
```

represents work-like accumulation along the curve and depends on orientation.

### Fundamental theorem for line integrals

If `F=∇φ` on an appropriate domain, then

```text
∫_C F·dr = φ(B)-φ(A).
```

The integral is path independent and every closed-loop integral is zero.

### Green's theorem

For a positively oriented simple closed curve bounding a suitable planar region `D`,

```text
∮_C P dx + Q dy
 = ∬_D (∂Q/∂x - ∂P/∂y) dA.
```

Green's theorem links circulation around a boundary with a derivative field over the enclosed region.

## Unit 7: Surface Integrals

### Definition of a surface integral

For a scalar field over a parametrized surface `S`,

```text
∬_S f dS
```

uses the surface-area factor derived from the cross product of the parameter tangent vectors.

### Applications

Surface integrals calculate area, distributed mass, flux, and related quantities on curved surfaces.

### Integrals over oriented surfaces

For an oriented surface with unit normal `n`, the flux of a vector field is

```text
∬_S F·n dS.
```

Reversing orientation changes the sign.

### Stokes' theorem

For a suitable oriented surface `S` with boundary `∂S`,

```text
∮_(∂S) F·dr = ∬_S (∇×F)·n dS.
```

It generalizes the planar circulation relation to surfaces in three dimensions.

### Divergence theorem

For a suitable volume `V` with outward-oriented boundary surface `∂V`,

```text
∯_(∂V) F·n dS = ∭_V ∇·F dV.
```

It relates total outward flux to the divergence throughout the enclosed volume.

## General Conceptual Framework

The course moves from one-dimensional limiting processes to multivariable and vector calculus:

```text
Sequences
   ↓
Infinite series
   ↓
Functions of several variables
   ↓
Multiple integrals
   ↓
Vector fields
   ↓
Line and surface integrals
   ↓
Green / Stokes / divergence theorems
```

Sequences and series formalize infinite approximation. Multivariable derivatives describe local change in several directions. Multiple integrals accumulate over areas and volumes. Vector-calculus theorems connect local derivatives inside a region with integrals over its boundary.

## Conceptual Distinctions

**Sequence ≠ series.** A sequence is an ordered list; a series is the limit problem associated with its partial sums.

**a_n→0 ≠ series convergence.** Vanishing terms are necessary but not sufficient for `Σa_n` to converge.

**Absolute convergence ≠ conditional convergence.** Absolute convergence permits safe rearrangement; conditional convergence can change under rearrangement.

**Agreement along a few paths ≠ proof of a multivariable limit.** One differing path disproves the limit, but finitely many agreeing paths do not establish it.

**Partial derivatives existing ≠ differentiability.** Differentiability requires a genuine first-order linear approximation.

**Gradient ≠ directional derivative.** The gradient is a vector; the directional derivative is its scalar projection onto a selected unit direction.

**Jacobian ≠ arbitrary correction factor.** It is the determinant that measures local coordinate scaling in a change of variables.

**Double integral ≠ repeated notation only.** Its domain is a two-dimensional region and the integration order reflects a description of that region.

**Line integral of a scalar field ≠ vector-field work integral.** `∫f ds` uses arc length; `∫F·dr` depends on field direction and curve orientation.

**Conservative field ≠ arbitrary field.** Under appropriate regularity and simply connected domain conditions, `∂P/∂y = ∂Q/∂x` in the plane is a fundamental criterion for conservativeness and path independence.

**Stokes' theorem ≠ divergence theorem.** Stokes relates boundary circulation to curl over a surface; divergence theorem relates outward flux to divergence over a volume.

**Boundary integral ≠ interior integral, but they can be linked.** Green, Stokes, and divergence theorems are different dimensional forms of the same boundary-versus-interior principle.

## References

- Ahmet Yesevi Üniversitesi Bilgisayar Mühendisliği Bölümü. *Matematik II* (TBIL102) ders materyalleri.
- Joel Hass, Christopher Heil, Maurice D. Weir, Przemyslaw Bogacki. *Thomas' Calculus*, 15th Edition. Pearson, 2022. ([Pearson](https://www.pearson.com/en-us/subject-catalog/p/thomas-calculus/P200000007103))
- Tom M. Apostol. *Calculus, Volume II*, 2nd Edition. Wiley, 1969.
- Walter Rudin. *Principles of Mathematical Analysis*, 3rd Edition. McGraw-Hill, 1976.
- Jerrold E. Marsden, Anthony J. Tromba. *Vector Calculus*, 6th Edition. W. H. Freeman, 2012.
- Michael Spivak. *Calculus on Manifolds*. W. A. Benjamin, 1965.

## Cite This Work

Köker, M. A. (2014). Calculus II: Sequences, Series and Multivariable Analysis. alikoker.com.tr. https://alikoker.com.tr/en/calculus-ii-sequences-series-multivariable-analysis

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