Consider the series ∑∞ n=0 √ 12(−1)n 3 n(2n + 1). Note: this was changed from ∑∞ n=1 √ 12(−1)n 3 n(2n + 1). (a) (3 marks

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Consider the series ∑∞
n=0
√
12(−1)n
3
n(2n + 1). Note: this was changed from ∑∞
n=1
√
12(−1)n
3
n(2n + 1).
(a) (3 marks) Use any test for convergence/divergence to show that
the series converges.
(b) (2 marks) It is possible to show that the sum of the series
∑∞
n=0
√
12(−1)n
3
n(2n + 1) is π, in other words, the series
converges to the number π. (You do NOT need to prove this, but it
can be done somewhat easily using a
Taylor series expansion of arctan x.)
Suppose you want to use a partial sum of this series to estimate
the value of π to an accuracy of within
0.0001. Would using the first 8 terms of the series be enough to
ensure you get an accuracy of within
0.0001? (8 terms means the terms where n = 0, 1, 2, 3, ....,
7.)