x2k+1 Exercise 1. Consider the Taylor series: tan -1 x = (-1)* with R = 1....

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x2k+1 Exercise 1. Consider the Taylor series: tan -1 x = (-1)* with R = 1. k=0 (a) Show that the above Taylor series converges for x = 1. 2k +1 T (b) Observe that = tan (1). Find a series for 4 4 T (c) Approximate - by adding the first 4 terms of the above series (use a calculator). Estimate 4 the error for this approximation. (d) Given that curate to? T = 4 0.785398163..., how many decimal places is the above approximation ac- (e) How many terms do we need to add if we want the approximation accurate to 3 decimal places? (f) How many terms do we need to add if we want the approximation accurate to 9 decimal places? (so that we get the above approximation). x2k+1 Exercise 1. Consider the Taylor series: tan -1 x = (-1)* with R = 1. k=0 (a) Show that the above Taylor series converges for x = 1. 2k +1 T (b) Observe that = tan (1). Find a series for 4 4 T (c) Approximate - by adding the first 4 terms of the above series (use a calculator). Estimate 4 the error for this approximation. (d) Given that curate to? T = 4 0.785398163..., how many decimal places is the above approximation ac- (e) How many terms do we need to add if we want the approximation accurate to 3 decimal places? (f) How many terms do we need to add if we want the approximation accurate to 9 decimal places? (so that we get the above approximation).

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Lets break this down step by step using the given information Given Taylor series arctan x sumk0infty frac1kx2k12k1 with R 1 where R is the radius of ... View the full answer