This Proof is right about sums and limits?

In summary, the conversation discusses a mathematical equation and its solution for an infinite number of terms. The conversation presents a proof using the sum of a geometric progression formula and discusses the use of infinite values. It also includes an image of the proof and a question about understanding the steps in the proof. The expert concludes by summarizing the key points and thanking the person for their help.
  • #1
MAGNIBORO
106
26
hello, sorry for bad English, i have a question.
if we consider the following equations and we take natural values note that tend 2
x-1=0 -----------------> x = 1
x^2-x-1=0 -----------------> x = 1.618033988 (golden ratio)
x^3-x^2-x-1=0 -----------------> x = 1.839286755
x^4-x^3-x^2-x-1=0 -----------------> x = 1.927561975
x^5-x^4-x^3-x^2-x-1=0 -----------------> x = 1.965948236
x^6-x^5-x^4-x^3-x^2-x-1=0 -----------------> x = 1.983582843

then we can assume that when the equation has infinite terms the answer is 2.
and reaches the following "proof" and let me know if it's right:
1 image:https://gyazo.com/49a46e56fb19b4ec7aa21594e4e78cd1
2 image:https://gyazo.com/c6b99485e6d0271c1c0bbdbaaca29d54

Besides knowing if this is OK too I wonder if this is what is called "inductive method"

thanks.
 
Last edited by a moderator:
Mathematics news on Phys.org
  • #2
No, you have a divergent series.
 
  • #3
I don't like your proof! If you have
x^n = 1 + x+ x^2 +...+x^(n-1)
using the formula for the sum of a GP you get
x^n = (1-x^n)/(1-x), and so
x - 2 + 1/x^n =0
now you can see that as n tends to infinity x gets as close as you like to 2.
 
  • #4
micromass said:
No, you have a divergent series.
I kept working and I think it gets to the real proof
https://gyazo.com/aeee169696eebe6c4357520f9dbaa837
 
Last edited by a moderator:
  • #5
You cannot use values like ##\infty## or ##\infty-2## like this. Ever.
 
  • #6
davidmoore63@y said:
I don't like your proof! If you have
x^n = 1 + x+ x^2 +...+x^(n-1)
using the formula for the sum of a GP you get
x^n = (1-x^n)/(1-x), and so
x - 2 + 1/x^n =0
now you can see that as n tends to infinity x gets as close as you like to 2.
I have a question.
How did you get in here
x^n = (1-x^n)/(1-x)
to here
x - 2 + 1/x^n =0
my poor mind can not compute
 
  • #7
micromass said:
You cannot use values like ##\infty## or ##\infty-2## like this. Ever.
okay and this?
https://gyazo.com/39a967c89bee8109366e6f62991acf26
 
Last edited by a moderator:
  • #8
MAGNIBORO said:
I have a question.
How did you get in here
x^n = (1-x^n)/(1-x)
to here
x - 2 + 1/x^n =0
my poor mind can not compute

Divide both sides by ##x^n## to get
## 1= \frac{x^{-n} - 1}{1-x}##
Multiply both sides by 1-x
##1 - x = x^{-n} -1##
Move everything to left
##0 = x^{-n} - 2 +x ##
 
  • #9
pwsnafu said:
Divide both sides by ##x^n## to get
## 1= \frac{x^{-n} - 1}{1-x}##
Multiply both sides by 1-x
##1 - x = x^{-n} -1##
Move everything to left
##0 = x^{-n} - 2 +x ##
thank you very much
 

Related to This Proof is right about sums and limits?

What is a limit in mathematics?

A limit in mathematics refers to the value that a function approaches as the input variable gets closer and closer to a specific value. It can also be thought of as the value that a function "approaches" but may never reach.

What is the purpose of using limits in mathematical proofs?

Limits are used in mathematical proofs to show that a certain value or function is true for all values or inputs, even if it may not be true at a specific point. It provides a way to generalize a result and prove it for infinite cases.

How do you find the limit of a function?

To find the limit of a function, you can use various techniques such as direct substitution, factoring, or using L'Hospital's rule. However, the most important step is to first determine the type of limit (finite, infinite, or oscillating) and then apply the appropriate method.

What are the common types of limits used in mathematical proofs?

The common types of limits used in mathematical proofs include one-sided limits, infinite limits, and limits at infinity. One-sided limits are used when the function approaches a different value from the left and right sides. Infinite limits are used when the function approaches positive or negative infinity. Limits at infinity are used when the function approaches a specific value as the input variable grows or decreases without bound.

What are common mistakes to avoid when using limits in proofs?

Common mistakes to avoid when using limits in proofs include assuming that a limit exists without properly proving it, confusing one-sided limits with two-sided limits, and not considering the behavior of the function at the point of the limit. It is important to carefully analyze the function and use the appropriate techniques to correctly evaluate the limit.

Similar threads

I A doubt about the multiplicity of polynomials in two variables
    • General Math
Replies
8
Views
1K
I Transformation of Functions: How Do Domain and Range Change?
    • General Math
Replies
4
Views
967
B Proof of Chain Rule: Understanding the Limits
    • General Math
Replies
3
Views
839
I Alternating Harmonic Numbers are cool, spread the word!
    • General Math
Replies
4
Views
561
I Time evolution of the electromagnetic wavefunction on a lattice
    • General Math
Replies
9
Views
1K
I What is the missing number in this number sequence pattern?
    • General Math
Replies
3
Views
634
B Popularizing a property for n-bonacci numbers without publishing it?
    • General Math
Replies
12
Views
980
MHB Interpolating Points with Continuous Modular Functions?
    • General Math
Replies
5
Views
882
A Rich "isotropic tensor" concept
    • General Math
Replies
1
Views
1K
MHB Fixed point,, Jacobi- & Newton Method, Linear Systems
    • General Math
Replies
7
Views
1K

Hot Threads

Recent Insights

Back
Top