lean_workbook_plus_52754

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Let $S=\frac15+\frac2{5^2}+\frac3{5^3}+\frac4{5^4}+\cdots$ ; then, \n \n \begin{align} 5S&=1+\frac2{5}+\frac3{5^2}+\frac4{5^3}+\cdots \ \phantom{\left(\frac11\right)^1}S&=0+\frac15+\frac2{5^2}+\frac3{5^3}+\cdots \ 4S&=1+\frac15+\left(\frac15\right)^2+\left(\frac15\right)^3+\left(\frac15\right)^4+\cdots \ \frac45S&=\frac15+\left(\frac15\right)^2+\left(\frac15\right)^3+\left(\frac15\right)^4+\cdots \ \frac{16}5S&=1 \ S&=\frac5{16} \end{align}

Why this node was retired

The posted statement is

theorem lean_workbook_plus_52754 (s : ℝ) (hs : s = ∑' n : ℕ, (n + 1) / (5 ^ (n + 1))) : s = 5 / 16   :=  by sorry

The intended real/rational quotients are computed in ℕ or ℤ (sometimes the cast occurs only after the sum), changing the expression.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Make every source fraction and summand rational or real before division, with casts on the bound index inside the summand; restore the source task: Let $S=\frac15+\frac2{5^2}+\frac3{5^3}+\frac4{5^4}+\cdots$ ; then, \n \n \begin{align} 5S&=1+\frac2{5}+\frac3{5^2}+\frac4{5^3}+\cdots \ \phantom{\left(\frac11\right)^1}S&=0+\frac15+\frac2{5^2}+\frac3{5^3}+\cdots \ 4S&=1+\frac15+\left(\frac15\right)^2+\left(\frac15\right)^3+\left(\frac15\right)^4+\cdots \ \frac45S&=\frac15+\left(\frac15\right)^2+\left(\frac15\right)^3+\left(\frac15\right)^4+\cdots \ \frac{16}5S&=1 \ S&=\frac5{16} \end{align}

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.