Question

In: Advanced Math

Compute the Taylor series at x = 0 for ln(1+x) and for x cos x by...

Compute the Taylor series at x = 0 for ln(1+x) and for x cos x by repeatedly differentiating the function. Find the radii of convergence of the associated series.

Expert Solution

Taylor series at x = 0 is called Maclaurin series.

Formula is,

For

So on.

So we can write

For

So on

So we can write

Now we need to find the radii of convergence of these series.

For that we can use ratio test.

then radius of convergence is

For first function

So on applying limits we get

To find convergence of we need to ratio test,

Series is convergent with radius of convergence equal to infinity

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

PLEASE GIVE AN UPVOTE!!!

Colby Messinger answered 1 year ago

Find the Taylor series for f(x)= ln(2+x) centered at x=0 and compute its interval of convergence....

Find the Taylor series for f(x)= ln(2+x) centered at x=0 and compute its interval of convergence. Show all work and use proper notation.

Write the Taylor series for f(x) = cos x about x = 0.

P1. Write the Taylor series for f(x) = cos x about x = 0. State the Taylor polynomials T2(x), T4(x), and T6(x) (note that T3(x) will be the same as T2(x), and T5(x) will be the same as T4(x)). Plot f(x), T2(x), T4(x), and T6(x), together on one graph, using demos or similar (cut-and-paste or reproduce below).

1. If f(x) = ln(x/4) -(a) Compute Taylor series for f at c = 4 -(b)...

1. If f(x) = ln(x/4) -(a) Compute Taylor series for f at c = 4 -(b) Use Taylor series truncated after n-th term to compute f(8/3) for n = 1,.....5 -(c) Compare the values from above with the values of f(8/3) and plot the errors as a function of n -(d) Show that Taylor series for f(x) = ln(x/4) at c = 4 represents the function f for x element [4,5]

Find the Taylor series of f(x)=ln(3+2x) about x=0.

Compute the Taylor polynomial indicated. f(x) = cos(x), a = 0 T5(x) = Use the error...

Compute the Taylor polynomial indicated. f(x) = cos(x), a = 0 T5(x) = Use the error bound to find the maximum possible size of the error. Round your answer to nine decimal places. cos(0.4) − T5(0.4) ≤

(a) Determine the Taylor Series centered at a = 1 for the function f(x) = ln...

(a) Determine the Taylor Series centered at a = 1 for the function f(x) = ln x. (b) Determine the interval of convergence for this Taylor Series. (c) Determine the number n of terms required to estimate the value of ln(2) to within Epsilon = 0.0001. Can you please help me solve it step by step.

Find the first 6 terms of the Taylor Series for ln (x) around 1, that is...

Find the first 6 terms of the Taylor Series for ln (x) around 1, that is x = c = 1. Then use your results to calculate ln (3.52) and get the absolute error. I'm supposed to find the first 6 terms of the Taylor series for the functions ln (x ) centered at 1 that is x = c = 1.( or a=1). Then use the results to calculate ln (3.52) and get the absolute error.

Recall that the following Taylor series is used to approximate Cosine: cos(x) = ∑ (−1) nx...

Recall that the following Taylor series is used to approximate Cosine: cos(x) = ∑ (−1) nx 2n (2n)! ∞ n=0 You have been tasked with developing an m-file that allows other engineers to quickly determine the minimum n needed to reduce the truncation error below an error threshold. The truncation error is the absolute error between the approximation and MATLAB’s cos() function. You may want to Google the Taylor series to better your understanding of it. Your code should perform...

Find the second-degree Taylor polynomial of f(x)=ln(1+sinx) centered at x=0.

find the n-th order Taylor polynomial and the remainder for f(x)=ln(1+x) at 0