Name: Name: University ID: Thomas Edison State University Calculus II (MAT-232) Section no.: Semester and year: Written Assignment 5 Answer all assigned exercises, and show all work. Each exercise is...

Name:
Name:
University ID:
Thomas Edison State University
Calculus II (MAT-232)
Section no.:
Semester and year:
Written Assignment 5
Answer all assigned exercises, and show all work. Each exercise is worth 5 points.
*Submitting a graph is not required; however, you are encouraged to create one for your own benefit and to include (or describe) one if possible.
Section 8.6
4. Determine the radius and interval of convergence.
Ans: We know,
Similarly we can also write, .
Now, differentiating both sides with respect to z0, we get,
Multiplying both sides by z0, we obtain,
Replacing, z0 by (x/2) we have
So, radius of convergence = 2, Interval of convergence = (-2,+2)
10. Determine the radius and interval of convergence.
    
Ans: Splitting the above series into two intervals and substracting we obtain,
Now, is re written as
That implies,
Radius of convergence = , interval of convergence =
12. Determine the radius and interval of convergence.

    
Ans: Matching the given series with the form,
In the given problem, we may write it as:
, where
For radius of convergence, we know,
Radius of convergence = , Interval of convergence =
16. Determine the radius and interval of convergence.

    
Ans: The radius and interval of convergence can be found using ratio test. By ration test if
So,
Canceling common terms we have,
Now for the series to be convergent,
, radius of convergence = 2 , Interval of convergence =
24. Determine the interval of convergence and the function to which the given power series converges.
    
    
According to ratio test,
Radius of convergence= 4, Interval of convergence=
26. Find a power series representation of f(x) about c = 0 (refer to example 6.6). Also, determine the radius and interval of convergence, and graph f(x) together with the partial sums and .
    
    
    
Ans: We know,
. For any values of , the series diverges. Now matching the given function with the expression above, we have a =3, r=x.
So,
, hence the power series is found out as f(x)
According to ratio test
Radius of convergence is , Interval of convergence =
28. Find a power series representation of f(x) about c = 0 (refer to example 6.6). Also, determine the radius and interval of convergence, and graph f(x) together with the partial sums and .
    
    
    
Ans:
So g(x) can be considered as the power series.
Radius of convergence is 1. Interval of convergence =
Section 8.7
4. Find the Maclaurin series (i.e., Taylor series about c = 0) and its interval of convergence.
    
    
Ans: Maclauric series can be written as
Now,
is divergent for all values of x, because .
Radius = , Interval =
6. Find the Maclaurin series (i.e., Taylor series about c = 0) and its interval of convergence.
    
    
Ans:
Now by ratio theory,
The series is convergent.
Radius = , Interval =
10. Find the Taylor series about the indicated center, and determine the interval of convergence.
    
    
Ans: A taylor series about a point is written as
Now,
By ratio test,
The series converges.
Radius = , Interval =
14. Find the Taylor series about the indicated center, and determine the interval of convergence.
    
    
Ans. Series about c=0 is similar to mclaurin series.
By ratio test:
Radius = 5, Interval =
22. Prove that the Taylor series converges to f(x) by showing that .
    
    
Ans:
The series cos (x) can be written as
We need to show that Rn tends to zero when x tends to infinity.
Here we will use the Lagrangian remainder concept about c=0,
But we have already seen that,
24. Prove that the Taylor series converges to f(x) by showing that .
    
    
Ans:
The remainder can be calculated as:
for finding the remainder let us consider the series around c=0,
But we know,
30. Use a known Taylor series to find the Taylor series about c = 0 for the given function, and find its radius of convergence.
    
    
Ans: Taylor series about c=0 is also similar to Mclaurin series.
Now to test convergence we perform ratio test.
By ratio test,
The series converges....