Q1.

Prove that

as

Show that the series does not converge.

Solution:

=

=

₀^∞

Put (n + 1) = - m

So,

₀^∞

Now,

=

= [where M is a huge number]

This is an alternating series almost close to 1.

Now we test the convergence of the series.

Let Be the m^th term of the series.

=

So, by ratio test, the series is not convergent.

Q2.

Obtain a one-term approximation for

I(x) =

as .

Solution:

I(x) =

=

=

=

=

[Let

]

=

=

= ₀^∞

For n = 0, the first term of the series is 2.

Q3.

Obtain two-term approximation for to the Laplace’s integral

As in the standard case - a problem we started in class. What is the leading-order approximation in the case f(c) = 0? Explain the error made in the intuitive calculation made in class.

Solution:

The Laplace’s integral is

The Laplace transform existence theorem states that if f(t) is piecewise continuous on every finite interval in [0, ∞] satisfying |f(t)|

_a^b

The two approximations of I(x) as can be written as

At x = c, f(c) = 0

The leading approximation =

As the other terms like becomes smaller.

The error made when we put |f(t)|.

Q4.

Obtain a two-term approximation for the Laplace’s integral as , for the maximum of ϕ is realized at an endpoint, say a, with ϕ.

Solution:

The Laplace integral is

As from Q3., we have

I(x)

Also, for the case where the maximum of is realized at an endpoint, say a, with ϕ.

Two term approximation

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Answer rating: 100% (QA)

Prove that as Show that the series does not converge Solution 0 Put n 1 m So 0 Now where M is a huge ... View the full answer