Power series in real world situation

In summary, the conversation discusses finding the required force to hold a weight in equilibrium at a given distance, using the equations Tcosθ=W and Tsinθ=F. It also mentions finding the force as a power series of θ and as a series of powers in x/l, where l is the length of the cable, W is the weight, and T is the tension in the cable. Ultimately, the goal is to expand F/W as a Taylor series.
  • #1
seboastien
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Homework Statement


A heavy weight is suspended by a cable and pulled to one side by a force F. How much force is required to hold the weight in equilibrium at a given distance x to one side. Tcosθ=W and Tsinθ=F. Find F/W as a power series of θ. Often in a problem like this, what we know is not θ, but x and l. Find F/W as a series of powers in x/l.

l is the length if the cable
W is weight
T is the tension in the cable


Homework Equations





The Attempt at a Solution



no idea...
 
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  • #2
The first obvious thing to do is to see that F/W= (T sin(θ))/(Tcos(θ))= tan(θ).

Can you expand that as a power series- say, a Taylor series?
 

FAQ: Power series in real world situation

What is a power series and how is it used in real world situations?

A power series is an infinite series of the form ∑n=0 an(x-c)n, where an are coefficients, x is the variable, and c is a constant. It is used to approximate functions in real world situations, such as in physics, engineering, and finance.

How do you determine the convergence of a power series?

The convergence of a power series can be determined by using the ratio test, where the limit of |an+1/an| as n approaches infinity is calculated. If this limit is less than 1, the series converges. Additionally, the ratio test can also be used to determine the radius of convergence, which is the range of values for x where the series converges.

3. Can power series be used to solve differential equations?

Yes, power series can be used to solve differential equations by representing the unknown function as a power series and then solving for the coefficients. This technique is known as the method of Frobenius and is commonly used in physics and engineering to solve differential equations in real world situations.

4. Are there any real world applications of Taylor series?

Yes, Taylor series, which are a special case of power series, are used in various real world applications. For example, they are used in numerical analysis for approximating functions, in physics for solving differential equations, and in finance for pricing financial instruments.

5. How do you find the sum of a power series?

The sum of a power series can be found by using the formula ∑n=0 an(x-c)n = a0 + a1(x-c) + a2(x-c)2 + ... + an(x-c)n + ..., where c is the center of the series. This formula can be used to find the sum of the series for a specific value of x, or it can be used to find the general form of the series by substituting in values for x and solving for the coefficients an.

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