Q9 (Approximation of π) (a) Show that 1/1 + t2 = 1 − t2 + t4 − ... + (−1)n−1 t 2n−2 + (−1)n t2n /1 + t2 for all t ∈ R and n ∈ N. (b) Integrate both side in (a), show that tan−1 (x) = x − x3/3 + x5 /5...

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Q9 (Approximation of π) (a) Show that 1 t2 1 +t2 for all t R and nEN (b) Integrate both side in (a), show that (-1)-112n-1 Q9 (Approximation of π) (a) Show that 1/1 + t² = 1 − t² + t⁴ − ... + (−1)ⁿ⁻¹ t ²ⁿ⁻² + (−1)ⁿ t²ⁿ /1 + t² for all t ∈ R and n ∈ N.

(b) Integrate both side in (a), show that tan⁻¹ (x) = x − x³/3 + x⁵ /5 − ... + (−1)ⁿ⁻¹x ²ⁿ⁻¹/ 2n − 1 + Z x 0 (−1)ⁿ t²ⁿ /1 + t² dt.

(c) Show that tan⁻¹ (x) − ( x − x³ /3 + x⁵ /5 − ... + (−1)ⁿ⁻¹x ²ⁿ⁻¹ /2n − 1) ≤ x ²ⁿ⁺¹ /2n + 1 .

(d) Show that tan⁻¹ 1/2 + tan⁻¹ 1/3 = π/4

(e) Show that π/4 − ((1/2 + 1/3 ) − 1/3(1/2³ + 1/3³ ) + 1/5 (1/2⁵ + 1/3⁵ ) − ... + (−1)ⁿ⁻¹ /2n − 1 ( 1/2 ²ⁿ⁻¹ + 1/ 3²ⁿ⁻¹ )) ≤ 1 /n · 2 ²ⁿ⁻¹

math Advanced-Math

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Answer 1

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Answer 2

Q9 (Approximation of π) (a) Show that 1/1 + t2 = 1 − t2 + t4 − ... + (−1)n−1 t 2n−2 + (−1)n t2n /1 + t2 for all t ∈ R and n ∈ N. (b) Integrate both side in (a), show that tan−1 (x) = x − x3/3 + x5 /5...

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Q9 (Approximation of π) (a) Show that 1/1 + t2 = 1 − t2 + t4 − ... + (−1)n−1 t 2n−2 + (−1)n t2n /1 + t2 for all t ∈ R and n ∈ N. (b) Integrate both side in (a), show that tan−1 (x) = x − x3/3 + x5 /5...

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Q9 (Approximation of π) (a) Show that 1/1 + t2 = 1 − t2 + t4 − ... + (−1)n−1 t 2n−2 + (−1)n t2n /1 + t2 for all t ∈ R and n ∈ N. (b) Integrate both side in (a), show that tan−1 (x) = x − x3/3 + x5 /5...

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Add Answer to: Q9 (Approximation of π) (a) Show that 1/1 + t2 = 1 − t2 + t4 − ... + (−1)n−1 t 2n−2 + (−1)n t2n /1 + t2 for all t ∈ R and n ∈ N. (b) Integrate both side in (a), show that tan−1 (x) = x − x3/3 + x5 /5...

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Add Answer to:
Q9 (Approximation of π) (a) Show that 1/1 + t2 = 1 − t2 + t4 − ... + (−1)n−1 t 2n−2 + (−1)n t2n /1 + t2 for all t ∈ R and n ∈ N. (b) Integrate both side in (a), show that tan−1 (x) = x − x3/3 + x5 /5...