a) Show that this series converges when x = 1. (You will need to use one of our tests.) (b) In fact, it can be shown that this series converges to arctan x when −1 ≤ x ≤ 1, so in particular, the series for x = 1 converges to arctan 1 = π/4. Hence if you multiply the above series by 4, you will have a series that converges to π. Write out this series for 4 arctan x, then use this fact and the first 10 nonzero terms of this series for 4 arctan x to find an approximation to π. Based on what you know the value of π to be, how many digits are correct in this approximation? (c) In an alternating series, the error in using the first n terms is always less than the absolute value of the next (i.e, the (n + 1)st) term. Use only this fact (i.e., do not use the error approximation that we talked about in class) to find the smallest value of n for which the error in using the above series for 4 arctan x to
1. Recall that the Taylor series for arctan x, expanded around x = 0, is given by
(a) Show that this series converges when x = 1. (You will need to use one of our
tests.)
(b) In fact, it can be shown that this series converges to arctan x when −1 ≤ x ≤ 1,
so in particular, the series for x = 1 converges to arctan 1 = π/4. Hence if you
multiply the above series by 4, you will have a series that converges to π. Write
out this series for 4 arctan x, then use this fact and the first 10 nonzero terms
of this series for 4 arctan x to find an approximation to π. Based on what you
know the value of π to be, how many digits are correct in this approximation?
(c) In an alternating series, the error in using the first n terms is always less than
the absolute value of the next (i.e, the (n + 1)st) term. Use only this fact (i.e.,
do not use the error approximation that we talked about in class) to find the
smallest value of n for which the error in using the above series for 4 arctan x to
approximate π is less than 1 × 10^−8
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