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Maybe this question is too dumb to be asked, but it's really bugging me so I decide to ask it anyway. I hope you bear with me.
Okay, it's known that both sides of the following series equal.
We all agree at this point. Now, each terms in $(1)$ and $(2)$ is a rational number. We all agree without a doubt. The sum of rational numbers is always a rational number. We agree again. Hence it follows that $\pi$ and $e$ must be rational numbers. However, it contradicts the well-known facts that both $\pi$ and $e$ are irrational numbers. So, where is my mistake?