On the decimal expansion of \prod_{k=1}^{\infty} \sum_{\ell =1}^{k} \frac{9}{10^\ell}

Philip Ramirez

Philip Ramirez

Answered question

2022-04-24

On the decimal expansion of
k=1=1k910

Answer & Explanation

silviavalls1005r

silviavalls1005r

Beginner2022-04-25Added 11 answers

Step 1
In general,
(q;q)=1qq2+q5+q7
which is explained in the Wikipedia articles Euler function, q-Pochhammer symbol, and Pentagonal number theorem. In your particular case, q=0.1 which explains the decimal expansion you are asking about since some powers of q are added and some subtracted.

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