Prove that sqrt 2 [ 1 - 1/3 + 1/5 - 1/7 +.]= pi 2 sqrt 2
Asked by Mr Beast
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1 Answer
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The expression inside the brackets looks like the Leibniz formula for? /4:
? /4 = 1 - 1/3 + 1/5 - 1/7 + .
So, the left side of the equation becomes:
?2 [? /4] = (?2)/4
Rationalize the denominator of the right side (? / (2?2):
(? / (2?2) * (?2 / ?2) = (?2) / (2 * 2) = (?2) / 4
Therefore?2 [1 - 1/3 + 1/5 - 1/7 + .] =? / (2?2) is true because both sides simplify to (?2) / 4
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