47 lines
1.5 KiB
Markdown
47 lines
1.5 KiB
Markdown
---
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id: 5900f4dd1000cf542c50ffef
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title: 'Problem 368: A Kempner-like series'
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challengeType: 5
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forumTopicId: 302029
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dashedName: problem-368-a-kempner-like-series
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---
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# --description--
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The harmonic series $1 + \\dfrac{1}{2} + \\dfrac{1}{3} + \\dfrac{1}{4} + ...$ is well known to be divergent.
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If we however omit from this series every term where the denominator has a 9 in it, the series remarkably enough converges to approximately 22.9206766193. This modified harmonic series is called the Kempner series.
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Let us now consider another modified harmonic series by omitting from the harmonic series every term where the denominator has 3 or more equal consecutive digits. One can verify that out of the first 1200 terms of the harmonic series, only 20 terms will be omitted. These 20 omitted terms are: $$\\dfrac{1}{111}, \\dfrac{1}{222}, \\dfrac{1}{333}, \\dfrac{1}{444}, \\dfrac{1}{555}, \\dfrac{1}{666}, \\dfrac{1}{777}, \\dfrac{1}{888}, \\dfrac{1}{999}, \\dfrac{1}{1000}, \\dfrac{1}{1110}, \\\\ \\dfrac{1}{1111}, \\dfrac{1}{1112}, \\dfrac{1}{1113}, \\dfrac{1}{1114}, \\dfrac{1}{1115}, \\dfrac{1}{1116}, \\dfrac{1}{1117}, \\dfrac{1}{1118}, \\dfrac{1}{1119}$$
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This series converges as well.
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Find the value the series converges to. Give your answer rounded to 10 digits behind the decimal point.
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# --hints--
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`euler368()` should return 253.6135092068.
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```js
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assert.strictEqual(euler368(), 253.6135092068);
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```
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# --seed--
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## --seed-contents--
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```js
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function euler368() {
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return true;
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}
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euler368();
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```
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# --solutions--
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```js
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// solution required
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```
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