What is the Origin of the Approximation?

In summary, the conversation is about a student trying to understand an approximation in a textbook derivation. The approximation is given as 1/a(1-x/a)^-1 ≈ 1/a(1+x/a), and the student is unsure of its origin. They are then told that it is a geometric series and the easiest of all.
  • #1
Beer-monster
296
0
Hi

Homework Statement



I'm trying to follow and work through a derivation in my textbook, making sure I can replicate the steps myself and understand what's happening. However, I came across this approximation and can't seem to figure out where it comes from or why and the book gives no clues.


Homework Equations



The general form is:

[tex]\frac{1}{a}\left(1-\frac{x}{a}\right)^{-1} \cong \frac{1}{a}\left(1+\frac{x}{a}\right) [/tex]


Could anyone please explain this to me?
 
Physics news on Phys.org
  • #2
1/(1-y)=1+y+y^2+y^3+... The series converges if y<1. Put y=(x/a) and approximate by only keeping the first two terms.
 
  • #3
Thanks a lot.

Could you tell me which series that is?
 
  • #4
Beer-monster said:
Thanks a lot.

Could you tell me which series that is?

Geometric. The easiest of all.
 
  • #5
I see it now.:blushing:

Been a while since I last saw one of those.
 

Related to What is the Origin of the Approximation?

What is the origin of approximation?

The origin of approximation can be traced back to ancient civilizations such as the Egyptians, Babylonians, and Greeks. These cultures used various methods of approximation in astronomy, mathematics, and engineering. However, the modern concept of approximation as a mathematical tool was developed in the 17th century by mathematicians such as Pierre de Fermat and Johannes Kepler.

Why is approximation important in science?

Approximation is important in science because it allows us to simplify complex systems or problems and make them more manageable. It also enables us to make predictions and draw conclusions even when we do not have exact information or measurements. In many cases, it is impossible to obtain exact solutions or measurements, so approximation provides a practical and useful approach.

What are some common methods of approximation?

There are several common methods of approximation used in science, including rounding, interpolation, extrapolation, and Taylor series. Rounding involves simplifying numbers by dropping decimal places or rounding to the nearest whole number. Interpolation is the process of estimating values between known data points. Extrapolation is a similar process, but it involves extending the estimation beyond the known data points. Taylor series is a mathematical method of approximating complex functions using a series of simpler functions.

What are the limitations of approximation?

While approximation is a useful tool, it also has its limitations. One of the main limitations is that it does not provide exact solutions or measurements, so there is always some level of error involved. Additionally, the accuracy of an approximation depends on the method used and the assumptions made. In some cases, the error introduced by approximation can be significant and lead to incorrect conclusions or predictions.

How does approximation relate to scientific modeling?

Approximation plays a critical role in scientific modeling. Models are simplified representations of complex systems or phenomena, and they often rely on approximation to make predictions or draw conclusions. Scientists use a combination of approximation methods and mathematical equations to develop models that can accurately represent real-world systems. The success of a model depends on the accuracy of the approximations used and the validity of the underlying assumptions.

Similar threads

Formula for Xo and Yo for graph of quadratic function
    • Precalculus Mathematics Homework Help
Replies
2
Views
900
I Landscape Fabric Roll
    • Calculus
Replies
8
Views
250
Basic coin toss problem to get 1 head, but with a different method
    • Precalculus Mathematics Homework Help
Replies
2
Views
852
Probability of getting 5 consecutive heads before 2 consecutive tails
    • Precalculus Mathematics Homework Help
Replies
12
Views
2K
B Approximation of the equation of ellipse
    • General Math
Replies
2
Views
899
I On error estimates of approximate solutions
    • Differential Equations
Replies
1
Views
688
Plotting the approximation of the Dirac delta function
    • Calculus and Beyond Homework Help
Replies
3
Views
414
How does this approximation work?
    • Calculus and Beyond Homework Help
Replies
22
Views
1K
I Exploring Continuous Approximation of 1D Random Walk Steps (Reif, pg 14)
    • Thermodynamics
Replies
1
Views
657
What is the relationship between random walks and destinations?
    • Precalculus Mathematics Homework Help
Replies
29
Views
1K

Hot Threads

Recent Insights

Back
Top