Is the statement 'int x^n from 0 to 1 = 1/(n + 1)' true?

  • Context: Undergrad 
  • Thread starter Thread starter cstvlr
  • Start date Start date
Click For Summary

Discussion Overview

The discussion revolves around the validity of the statement regarding the integral of x^n from 0 to 1, specifically whether it equals 1/(n + 1) and how this holds as n approaches infinity. The scope includes mathematical reasoning and exploration of conditions under which the statement is true or false.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • Some participants assert that the integral int x^n from 0 to 1 equals 1/(n + 1) but note that as n approaches infinity, the result tends towards zero.
  • Others point out that for values of x between 0 and 1, x^n decreases as n increases, suggesting that the integral's value diminishes accordingly.
  • One participant clarifies that the statement holds true for n > -1, while it does not hold for n < 0.
  • Another participant corrects the previous claim to specify that the statement is valid for n ≥ 0, while also noting that it is not true for n < 0.
  • Further clarification is provided that the statement is indeed true for n > -1, emphasizing the importance of considering non-integer values.

Areas of Agreement / Disagreement

Participants express differing views on the conditions under which the integral statement is true, with some agreeing on the range of n while others provide corrections and clarifications. The discussion remains unresolved regarding the precise conditions for validity.

Contextual Notes

There are limitations regarding the assumptions made about the values of n, particularly concerning negative values and whether the discussion applies to integers only or extends to real numbers.

cstvlr
Messages
12
Reaction score
0
int x^n from 0 to 1 = 1/(n + 1)

But as n approaches infinity the answer becomes zero.
 
Physics news on Phys.org
Yep. And if you look at x^n for some number 0<x<1 and 0<n, x^n will get smaller as n increases. It's only natural to see the integral get smaller as well.
 
Ok, thanks.
 
cstvlr said:
int x^n from 0 to 1 = 1/(n + 1)

But as n approaches infinity the answer becomes zero.

It is true as long as n &gt; -1. For n < 0, it is not true.
 
Last edited:
Mute said:
It is true as long as n \geq 0. For n < 0, it is not true.
Correction: It is true as long as n > -1, not zero.
 
D H said:
Correction: It is true as long as n > -1, not zero.

Corrected. Thanks. I forgot to edit that when I decided not to talk only about integers.
 

Similar threads

Undergrad Infinite Series - Divergence of 1/n question
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Find ##\int_0^π \sin^ n x dx ##
  • · Replies 5 ·
Replies
5
Views
5K
Graduate Limit of ##i^\frac{1}{n}## as ##n \to \infty##
  • · Replies 14 ·
Replies
14
Views
2K
Undergrad Solving the Difficult Integral ##\int_0^{\infty} x^{n+1} e^{-x} \sin(ax) dx##
  • · Replies 5 ·
Replies
5
Views
4K
Undergrad Integrating x^ne^xn: Analyzing & Solving
  • · Replies 5 ·
Replies
5
Views
4K
Undergrad Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
  • · Replies 9 ·
Replies
9
Views
3K
High School Difficulty with understanding whether 1/n converges or diverges
  • · Replies 11 ·
Replies
11
Views
6K
Undergrad Understanding Theorem 13 from Calculus 7th ed, R. Adams, C. Essex, 4.10
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Determination of error in interpolating polynomial
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Proof of 1/(x^2) Not Having Limit at 0
  • · Replies 5 ·
Replies
5
Views
2K