Use the power series for \(\displaystyle{\arctan{{\left({x}\right)}}}\) to

Aiyana Ayers

Aiyana Ayers

Answered question

2022-04-08

Use the power series for arctan(x) to derive an expression for π
Given that the power series for arctanx is given by
n=0(1)nx2n+12n+1
for x[1,1] how would you show that
π=23n=0(1)n(2n+1)3n?
I've tried by plugging in x=1 to get
π4=n=0(1)n(2n+1)
but other then plugging in values I have no clue how to reach the conclusion

Answer & Explanation

kinoNoidae1wj

kinoNoidae1wj

Beginner2022-04-09Added 7 answers

tanπ6=13
π=6arctan13
arctanx=n=0(1)nx2n+12n+1
π=6arctan13=6(1)n((13)12)2n+12n+1
π=633(1)n(13)n2n+1
π=23(1)n3n(2n+1)

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