0% Complete
0/59 Steps

Integral under limit: TIFR 2019 GS Part A, Problem 11

Understand the problem

Let \(f\) be a continuous function on \([0,1]\). Then the limit \(\lim_{n \to \infty} \int_0^1 nx^nf(x)dx\) is equal to
  • \(f(0)\)
  • \(f(1)\)
  • \(sup_{x\in [0,1]}f(x)\)
  • The limit does not exist.
Source of the problem
TIFR 2019 GS Part A, Problem 11
Topic
Analysis
Difficulty Level
Hard
Suggested Book
Real analysis, Bartle,Sherbert

Start with hints

Do you really need a hint? Try it first!

Consider \( g(t)=f(t)-f(1)\). Can you prove limit \(\lim_{n \to \infty}| \int_0^1 nx^ng(x)dx|=0\) ?
Try to bound \(g(t)\) in the neighbourhood of \(1\) then try to prove hint 1
\(| \int_0^1 nx^ng(x)dx|\leq |\int_{0}^{1-\delta}nx^ng(x)dx|+|\int_{1-\delta}^{1} nx^ng(x)dx|\)
Once you are done with hint 1, what can you say about option 2?
\(g(t) \to 0\) as \(t \to 1\).  
  1. So, for \(\epsilon >0,\exists \delta>0\) s.t \(|g(t)|<\epsilon, \forall |x-1|<\delta\)
\(| \int_0^1 nx^ng(x)dx|\leq |\int_{0}^{1-\delta}nx^ng(x)dx|+|\int_{1-\delta}^{1} nx^ng(x)dx| \leq \int_{0}^{1-\delta}nx^n|g(x)|dx+\int_{1-\delta}^{1} nx^n|g(x)|dx =I_1+I_2\). Now, \(I_1\to 0\) as \(n \to \infty\) as \(nx^n|g(x)|\) is uniformly convergent to zero as \(n \to \infty\) now \(I_2< \frac{n}{n+1}\epsilon\) because of point 1) so we are done!!    

Watch the video (Coming Soon)

Connected Program at Cheenta

College Mathematics Program

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

Similar Problems

Multiplicative group from fields: TIFR GS 2018 Part A Problem 17

This problem is a cute and simple application on the Multiplicative group from fields in the abstract algebra section. It appeared in TIFR GS 2018.

Group with Quotient : TIFR GS 2018 Part A Problem 16

This problem is a cute and simple application on Group theory in the abstract algebra section. It appeared in TIFR GS 2018.

Symmetric groups of order 30: TIFR GS 2018 Part A Problem 23

This problem is a cute and simple application on the Symmetric groups of order 30 in the abstract algebra section. It appeared in TIFR GS 2018.

Matrix additive-multiplicative :TIFR 2018 Part A Problem 19

This problem is a cute and simple application of additive and multiplicative properties of matrices in the linear algebra section. It appeared in TIFR GS 2018.

Commutative does not commute in matrices: TIFR 2018 Part A, Problem 11

This problem is a cute and simple approach using beautiful fact of matrices transformation in the linear algebra section. a, b and c can be taken from any commutative ring with identity, often taken to be the ring of real numbers or the ring of integers .It appeared in TIFR GS 2018.

Spanning matrix space by niltopent matrices: TIFR 2018 Part A, Problem 15

This problem is a cute and simple application of spanning matrix space by niltopent matrices in the linear algebra section. It appeared in TIFR GS 2018.

Calculating eigenvalues through geometry:TIFR 2018 Part A, Problem 14

This problem is a cute and simple application of calculating eigenvalues through geometry in the linear algebra section. It appeared in TIFR GS 2018.

Direct product of groups: TIFR 2018 Part A, Problem 9

This problem is a cute and simple application of the direct product of groups in the abstract algebra section. It appeared in TIFR GS 2018.

Real Symmetric matrix: TIFR 2018 Part A, Problem 6

This problem is a cute and simple application of the real symmeric matrices in the linear algebra section. It appeared in TIFR GS 2018.

Integration of nonnegative function: TIFR 2018 Part A, Problem 4

This problem is a cute and simple application of spanning matrix space by niltopent matrices in the linear algebra section. It appeared in TIFR GS 2018.