In expressions involving an infinite process (infinite sum, infinite sequence of nested radicals), sometimes the hardest part is proving the existence of a well-defined value. Consider, for example, Ramanujan's infinite nested radical: $$ \sqrt{1+2\sqrt{1+3\sqrt{1+\ldots}}}. \qquad(*) $$ Assuming the above is well-defined, there is a slick trick showing that it evaluates to $3$.

But such careless assumptions can lead to trouble, as in the example of the expression: $$ -5 + 2(-6 + 2(-7 + 2(-8 + \ldots))). \qquad(**) $$

Applying the identity $n = -(n + 2) + 2(n + 1)$ repeatedly for $n=3,4,5,\ldots$, we get \begin{align} 3 &= -5 + 2(4) \\ &= -5 + 2(-6 + 2(5))\\ &= -5 + 2(-6 + 2(-7 + 2(6))\\ &= -5 + 2(-6 + 2(-7 + 2(-8 + 2(7)))\\ &=\ldots, \end{align} which would falsely suggest that $(**)$ evaluates to $3$.

What are some interesting examples where evaluating an expression assuming its existence is much easier than proving existence?

Edit: clarifying in light of some of the discussion in the comments. I can see how $(**)$ can also invite examples of false conclusions from an assumption of existence. That was not the intent of the question; the sole point of $(**)$ was to show that the solution technique to $(*)$ provided at the link can in general yield false conclusions if existence is assumed without additional proof. The spirit of the question is to exhibit cases where the limit exists, but its value given the existence is much easier to establish than the existence itsef.

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