Chapter 23: Real Analysis and Series of Functions (Set-1)
Which set equals the real line ℝ?
A All rational numbers
B All complex numbers
C All real numbers
D Only integers
The real line ℝ contains every real number, including rationals and irrationals. It is represented as a continuous line with no gaps, unlike the rationals alone.
Which is an open interval?
A [a, b]
B [a, b)
C (a, b]
D (a, b)
In an open interval (a, b), endpoints a and b are not included. Points strictly between a and b belong to the set, matching the definition of open intervals.
Which is a closed interval?
A [a, b]
B (a, b)
C (a, b]
D [a, b)
A closed interval [a, b] includes both endpoints. It contains all x with a ≤ x ≤ b, and it is closed in the usual real-number topology.
What is a neighborhood of a real number a?
A Points exactly at a
B Points within some radius
C All rationals near a
D All integers near a
A neighborhood of a contains all real numbers within distance r from a for some r > 0, typically written (a − r, a + r). It captures “closeness” using distance.
Which expression defines absolute value |x|?
A Always x
B Distance from 0
C Distance from 1
D Always −x
|x| equals the distance of x from 0 on the real line. It is nonnegative and measures magnitude: |x| = x if x ≥ 0, and |x| = −x if x < 0.
What does “x ≤ y” mean on real line?
A x is left of y
B x equals y
C x is right of y
D x is not comparable
On the real line, ordering matches position: x ≤ y means x lies to the left of y or at the same point. This is the usual order relation on ℝ.
Which set property is unique to ℝ?
A Has only integers
B Only finite sets
C Only rationals
D Least upper bound
The completeness property of ℝ says every nonempty set bounded above has a least upper bound (supremum). This is not true in ℚ, making it a key feature of real numbers.
Supremum of a set is
A Any upper bound
B Greatest element only
C Least upper bound
D Least element only
The supremum (sup) is the smallest number that is still an upper bound. It may not belong to the set, but it is the tightest possible upper bound.
Infimum of a set is
A Greatest lower bound
B Any lower bound
C Greatest element
D Least upper bound
The infimum (inf) is the largest number that is still a lower bound. Like supremum, it might not be contained in the set, but it is the best lower bound.
If a set has a maximum, then its supremum is
A Smaller than maximum
B Larger than maximum
C Not defined
D Equal to maximum
When a set contains its greatest element (maximum), that element is also the least upper bound. So maximum and supremum coincide in this case.
Which set is bounded above?
A All integers
B (−∞, 5]
C ℝ itself
D (0, ∞)
(−∞, 5] has an upper bound 5 because every element is ≤ 5. Sets like ℝ or (0, ∞) have no finite upper bound and are unbounded above.
Which set is bounded?
A (0, 1)
B (0, ∞)
C (−∞, 3)
D All integers
A bounded set is bounded above and below. (0, 1) is contained between 0 and 1, so it is bounded. The others extend to infinity in at least one direction.
A set is bounded below if
A Has maximum element
B Has some lower bound
C Has supremum only
D Contains 0
Bounded below means there exists a number m such that m ≤ x for every x in the set. That m is called a lower bound; it need not belong to the set.
What is the usual metric on ℝ?
A d(x,y)=x+y
B d(x,y)=|x−y|
C d(x,y)=xy
D d(x,y)=x/y
The standard distance on real numbers is d(x, y) = |x − y|. It measures how far apart two real numbers are and satisfies metric properties like symmetry and triangle inequality.
Archimedean property means
A n exceeds any real
B ℝ has gaps
C Integers are bounded
D Only rationals exist
The Archimedean property states that for any real number x, there exists an integer n such that n > x. This prevents infinitely large “finite” real numbers.
Density of rationals means
A ℚ has maximum
B Between reals exists rational
C ℚ is uncountable
D ℚ equals ℝ
Between any two distinct real numbers a < b, there exists a rational number q with a < q < b. This is the density of ℚ inside ℝ.
Between any two distinct reals, there exists
A Only integers
B No irrational
C Only natural numbers
D A rational number
This follows from density of rationals. No matter how close two real numbers are, you can always find a rational number strictly between them.
Least upper bound property applies to
A Nonempty bounded-above set
B Empty set
C Any set of ℝ
D Only finite sets
The completeness axiom states: every nonempty subset of ℝ that is bounded above has a supremum in ℝ. Without boundedness or nonemptiness, it doesn’t apply.
If S is bounded below and nonempty, then
A sup S must be 0
B S must be finite
C S must contain inf
D inf S exists in ℝ
By completeness, a nonempty set bounded below has an infimum (greatest lower bound) in ℝ. The infimum may or may not be an actual element of S.
Which indicates a set is unbounded?
A Has upper bound
B No finite bound exists
C Has lower bound
D Contains a maximum
A set is unbounded if it is not bounded above or not bounded below. That means no single real number can serve as a finite bound in that direction.
Pointwise convergence means fₙ(x) → f(x)
A For each fixed x
B For all x, same speed
C Only at one point
D Only uniformly
Pointwise convergence means: for each fixed x, the sequence of numbers fₙ(x) approaches f(x) as n→∞. The speed of convergence may depend on x.
Uniform convergence means fₙ → f
A Depends on x
B Same control for all x
C Only at endpoints
D Only for series
Uniform convergence means one N works for all x to make |fₙ(x)−f(x)| small. Unlike pointwise convergence, the bound does not depend on x.
Which is stronger?
A Pointwise convergence
B Both same always
C Uniform convergence
D Neither implies other
Uniform convergence implies pointwise convergence, but not vice versa. Uniform convergence controls the error over the whole domain at once, so it is a stronger form.
Cauchy criterion for uniform convergence says
A Differences get small uniformly
B fₙ(x) equals f(x)
C Only derivatives converge
D Only integrals converge
A sequence (fₙ) is uniformly Cauchy if for every ε>0, there exists N such that for m,n≥N, |fₙ(x)−f_m(x)|<ε for all x. This matches uniform convergence in complete spaces.
If fₙ are continuous and converge uniformly, then f is
A Discontinuous always
B Continuous
C Undefined
D Only piecewise continuous
Uniform limit of continuous functions is continuous. Uniform convergence allows taking limits inside continuity arguments, preventing “limit breaks continuity” problems common in mere pointwise convergence.
Uniform convergence helps interchange limit and
A Random choice
B Continuity operations
C Only inequalities
D Only algebra
With uniform convergence, one can often pass the limit through continuous operations safely, such as evaluating continuity of the limit function and exchanging limits with integration under suitable conditions.
Weierstrass M-test is used for
A Pointwise divergence
B Uniform convergence of series
C Finding derivatives only
D Solving equations
The Weierstrass M-test gives a simple way to prove uniform convergence of a series of functions ∑ fₙ by comparing |fₙ(x)| with a convergent numeric series ∑ Mₙ.
M-test requires |fₙ(x)| ≤
A fₙ(x) itself
B A constant Mₙ
C x always
D 1/n always
If for all x, |fₙ(x)| ≤ Mₙ and ∑ Mₙ converges, then ∑ fₙ converges uniformly (and absolutely) on the domain. This is the Weierstrass M-test.
A series of functions is ∑ fₙ(x). It converges pointwise if
A ∑ fₙ(x) converges for each x
B Same N works all x
C fₙ are continuous
D Domain is bounded
Pointwise convergence of a function series means for each fixed x, the numeric series ∑ fₙ(x) converges. Uniform convergence adds a uniform control over x.
Uniform convergence of ∑ fₙ(x) means partial sums S_N(x)
A Converge uniformly
B Converge only at one x
C Diverge always
D Must be polynomials
A series of functions converges uniformly if its sequence of partial sums S_N(x) converges uniformly to a function S(x). This ensures stable behavior under limits across the domain.
A power series centered at a is
A ∑ aₙx
B ∑ aⁿxₙ
C ∑ (x+a)
D ∑ aₙ(x−a)ⁿ
A power series has the form ∑ aₙ(x−a)ⁿ, where a is the center. Its convergence depends on x through |x−a|, leading to a radius and interval of convergence.
Radius of convergence R describes
A Where it converges absolutely
B Where it always diverges
C Only endpoint behavior
D Only derivative existence
A power series converges absolutely when |x−a| < R and diverges when |x−a| > R. Endpoints |x−a|=R need separate testing and may converge or diverge.
Which test often finds radius R?
A Mean value theorem
B Ratio test
C Intermediate value theorem
D Fermat theorem
The ratio test commonly computes the limit of |aₙ₊₁/aₙ| to determine R for ∑ aₙ(x−a)ⁿ. It gives a clean condition for absolute convergence inside the radius.
Inside the radius, a power series converges
A Conditionally only
B Absolutely
C Never
D Only at center
For |x−a| < R, the series behaves like a convergent geometric-type comparison and converges absolutely. Absolute convergence inside the radius is a standard power series property.
Outside the radius, a power series
A Diverges
B Converges absolutely
C Always equals 0
D Converges at infinity
If |x−a| > R, terms do not go to zero fast enough and the series diverges. This is part of the fundamental “radius of convergence” theorem for power series.
At endpoints |x−a|=R, convergence is
A Always convergent
B Always divergent
C Must be checked separately
D Same as inside
At boundary points, the general radius theorem gives no final answer. Each endpoint requires separate tests (comparison, alternating, etc.), so one endpoint may converge and the other may diverge.
Term-by-term differentiation of a power series is valid
A Only at endpoints
B Inside radius
C Only at x=0
D Never valid
A power series can be differentiated term-by-term for all x with |x−a|
Term-by-term integration of a power series is valid
A Only outside radius
B Inside radius
C Only at x=a
D Only for finite sums
Inside the radius of convergence, power series can be integrated term-by-term. The integrated series also has the same radius of convergence, giving a reliable method to build new series.
Maclaurin series is Taylor series centered at
A a=0
B a=1
C a=−1
D a=π
Taylor series expands a function around x=a. When a=0, it is called the Maclaurin series. It expresses functions using powers of x with coefficients from derivatives at 0.
Power series are called “analytic” idea because they
A Always finite
B Represent functions locally
C Never converge
D Ignore derivatives
Many analytic functions can be represented by a power series around a point within some radius. This links smooth derivatives to series representation and makes local behavior computable.
A real sequence (xₙ) converges to L if
A xₙ equals L always
B xₙ gets arbitrarily close
C xₙ is bounded only
D xₙ is increasing only
Convergence means: for every ε>0, there exists N such that for n≥N, |xₙ−L|<ε. This formalizes “xₙ gets as close as desired” to L for large n.
A Cauchy sequence means
A Terms approach 0
B Terms get close to each other
C Terms are monotone
D Terms are periodic
A sequence is Cauchy if for every ε>0 there exists N such that m,n≥N implies |xₙ−x_m|<ε. In ℝ, every Cauchy sequence converges.
Completeness of ℝ implies
A Every bounded set has max
B Every Cauchy sequence converges
C Every sequence is monotone
D Every set is countable
Completeness means no “gaps” in ℝ. One key result is that every Cauchy sequence of real numbers converges to a real limit, which is not always true in ℚ.
Monotone convergence theorem (basic) says
A Bounded monotone sequence converges
B Any sequence converges
C Only decreasing converges
D Only increasing diverges
If a real sequence is monotone (nondecreasing or nonincreasing) and bounded, then it converges. The limit equals the supremum (increasing) or infimum (decreasing) of its set of values.
Bolzano–Weierstrass theorem states
A Every series converges
B Every bounded sequence has convergent subsequence
C Every function is continuous
D Every open set is closed
In ℝ, boundedness guarantees a convergent subsequence. This theorem is foundational for compactness ideas and helps prove many results about limits without requiring full sequence convergence.
A series ∑ aₙ converges absolutely if
A ∑ aₙ converges
B ∑ |aₙ| converges
C aₙ alternates
D aₙ decreases
Absolute convergence means the series of absolute values converges. This is stronger than conditional convergence and guarantees rearrangements do not change the sum, unlike conditionally convergent series.
Alternating series test needs terms that
A Decrease to 0
B Increase in size
C Stay constant
D Are all positive
For an alternating series ∑ (−1)ⁿbₙ with bₙ≥0, if bₙ decreases and bₙ→0, then the series converges. It’s a simple, common convergence test.
Comparison test compares ∑ aₙ with
A A function only
B Another known series
C A derivative only
D A random sequence
The comparison test uses a known benchmark series. If 0 ≤ aₙ ≤ bₙ and ∑ bₙ converges, then ∑ aₙ converges. If aₙ ≥ bₙ and ∑ bₙ diverges, ∑ aₙ diverges.
Root test checks limit of
A aₙ₊₁ − aₙ
B √[n]{|aₙ|}
C aₙ / n
D n·aₙ
The root test uses L = lim sup √[n]{|aₙ|}. If L<1, the series converges absolutely; if L>1, it diverges; if L=1, the test is inconclusive.
Uniform convergence is most useful because it allows
A Changing domain size
B Swapping limit with integration
C Making series finite
D Removing continuity
Under suitable conditions, uniform convergence helps justify interchanging limits with integrals (and often with continuity). This prevents incorrect steps that can happen with only pointwise convergence