Question

1, 2. It is probably evident that the Gregory/Leibniz series converges very slowly. The reason is that with the powers ofx in the Taylor series do not decrease in size. Here is an idea for obtaining better approximations. tan x+ tan y 1-tan rtan y a. The addition rule for the tangent says tan(x+y)- anpan qnAThe objective is to find small numbers p and q such thai can be done then we have tan-ı pHan-qu tan-,--, where a series can be used to approximate tan-P and tan q. Show that withpl/2 and q - 1/3, we have ^-tan . Let pe tan x and q-an y to show that 24모 l-pg 1 . If it 1-P9 b. Write the series for tan (1/2) and tan (l/3). What is the error in the approximation to x/4 using ten terms? (Assume a calculator gives the "exact" value ofr.) c. Use the Remainder in Alternating Series Theorem to estimate the error introduced when the series are terminated after 100 terms. d. How many terms of this series must be used to obtain an approximation to x with an error no greater than 10°? Compare this approximation to that of Step I for speed ofconvergence

2. It is probably evident that the Gregory/Leibniz series converges very slowly. The reason is that with x = 1, the powers of x in the Taylor series do not decrease in size. Here is an idea for obtaining better approximations.

I need help with d, please. Thanks in advance

Answer 1

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