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take one subtract a third add a fifth subtract the seventh add a ninth and so on forever alternating adding and subtracting one divided by all the odd numbers and you get this which happens to be equal to pi over four definitely a pretty sweet result though we wont dwell now on why pi shows up here it would be a tangent but now lets rearrange these terms a little bit to make a different looking infinite series take the first two positive terms one and one-fifth and then put the first negative term negative one-third after them then tack on the next two positive terms one-ninth and one-thirteenth and follow them both with the next negative term negative a seventh and keep going always adding the next two positive terms followed by the next single negative term in the pattern plus plus minus plus plus minus and so on and we get [Music] ew not the same thing wait what that cant be right this new infinite sum is just a rearrangement of the terms of the first infinite sum we didnt add
