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Chapter 1: Q- 26E (page 1)

Question: In Problems 23–26, express the given power series as a series with generic term.

26.∑n=1∞ann+3xn+3

Short Answer

Expert verified

The required term is ∑k=4∞ak-3kxk

Step by step solution

01

Power series.

A power series is an infinite series of the form,

∑n=0∞an(x-c)n=a0+a1(x-c)+a2(x-c)2+.....

Where,an represents the coefficient term of the nth term and c is a constant.

02

To express the given series in terms of the generic term

In order to express the given series in terms of generic term xk , we will change the index of the power series.

Given that, f(x)=∑n=1∞ann+3xn+3.

Let,

n+3=kn=k-3

So,

role="math" localid="1664278859581" ∑n=1∞ann+3xn+3=∑k-3=1∞ak-3kxk=∑k=4∞ak-3kxk

Hence, the required term is∑k=4∞ak-3kx.

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Most popular questions from this chapter

In Project C of Chapter 4, it was shown that the simple pendulum equationθ''(t)+sinθ(t)=0 has periodic solutions when the initial displacement and velocity show that the period of the solution may depend on the initial conditions by using the vectorized Runge–Kutta algorithm with h= 0.02 to approximate the solutions to the simple pendulum problem on

[0, 4] for the initial conditions:

localid="1664100454791" (a)  θ(0)=0.1,θ'(0)=0(b)  θ(0)=0.5,θ'(0)=0(c)  θ(0)=1.0,θ'(0)=0

[Hint: Approximate the length of time it takes to reach].

Use Euler’s method with step size h = 0.1 to approximate the solution to the initial value problem

y'=x-y2, y (1) = 0 at the points x=1.1,1.2,1.3,1.4and1.5.

In Problems 9–20, determine whether the equation is exact.

If it is, then solve it.

et(y - t)dt +(1 +et)dy = 0

Mixing.Suppose a brine containing 0.2 kg of salt per liter runs into a tank initially filled with 500 L of water containing 5 kg of salt. The brine enters the tank at a rate of 5 L/min. The mixture, kept uniform by stirring, is flowing out at the rate of 5 L/min (see Figure 2.6).

(a)Find the concentration, in kilograms per liter, of salt in the tank after 10 min. [Hint:LetAdenote the number of kilograms of salt in the tank attminutes after the process begins and use the fact that

rate of increase inA=rate of input- rate of exit.

A further discussion of mixing problems is given in Section 3.2.]

(b)After 10 min, a leak develops in the tank and an additional liter per minute of mixture flows out of the tank (see Figure 2.7). What will be the concentration, in kilograms per liter, of salt in the tank 20 min after the leak develops? [Hint:Use the method discussed in Problems 31 and 32.]

Nonlinear Spring.The Duffing equationy''+y+ry3=0 where ris a constant is a model for the vibrations of amass attached to a nonlinearspring. For this model, does the period of vibration vary as the parameter ris varied?

Does the period vary as the initial conditions are varied? [Hint:Use the vectorized Runge–Kutta algorithm with h= 0.1 to approximate the solutions for r= 1 and 2,

with initial conditionsy(0)=a,y'(0)=0 for a = 1, 2, and 3.]
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