2. The Taylor series about x = 0 for the arctangent function converges for 1 <...

  

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2. The Taylor series about x = 0 for the arctangent function converges for 1 < x 1 and is given by tan (x) - = 3-1/2 n=0 (1),2n+1 2n + 1 Write a computer program to evaluate tan-(x) by truncating the series to N + 1 terms. (a) Use your program to approximate by evaluating tan-(1) for different values of N. Give the approximations and errors in a table. To compute the errors, use that ~ 3.141592653589793. (b) Repeat part (a) using x = (c) Plot the errors in the approximations from parts (b) and (c) vs. the number of terms in the truncated sum on a log-log graph and a semilog (log-linear) graph (MATLAB commands are loglog and semilogy, respectively). What do these graphs indicate about the rate of convergence? (d) Analytically derive bounds for the errors in using the truncated series to approximate using the series from parts (a) and (b). 2. The Taylor series about x = 0 for the arctangent function converges for 1 < x 1 and is given by tan (x) - = 3-1/2 n=0 (1),2n+1 2n + 1 Write a computer program to evaluate tan-(x) by truncating the series to N + 1 terms. (a) Use your program to approximate by evaluating tan-(1) for different values of N. Give the approximations and errors in a table. To compute the errors, use that ~ 3.141592653589793. (b) Repeat part (a) using x = (c) Plot the errors in the approximations from parts (b) and (c) vs. the number of terms in the truncated sum on a log-log graph and a semilog (log-linear) graph (MATLAB commands are loglog and semilogy, respectively). What do these graphs indicate about the rate of convergence? (d) Analytically derive bounds for the errors in using the truncated series to approximate using the series from parts (a) and (b).

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To evaluate tan1x using the given Taylor series we can write a Python program as f... View the full answer