# Will this series converge (PI related) ?

**URL:** <https://boards.straightdope.com/t/will-this-series-converge-pi-related/715759>\
**Category:** Factual Questions\
**Created:** [March 23, 2015, 4:16am UTC](https://boards.straightdope.com/t/will-this-series-converge-pi-related/715759 "2015-03-23T04:16:05Z")\
**Posts on this page:** 4\
**Page:** 1

<div class="post-metadata">

**Author:** ![am77494](https://avatars.discourse-cdn.com/v4/letter/a/d2c977/32.png) [@am77494](https://boards.straightdope.com/u/am77494)\
**Post date:** [March 23, 2015, 4:16am UTC](https://boards.straightdope.com/t/will-this-series-converge-pi-related/715759/1 "2015-03-23T04:16:05Z")

</div>

I came across a good question - maybe it was answered before, if so please point me to it.

Consider the series 1/a\_n where a\_n = n th digit of pi. So, S(n), the sum of the series :

S(n)=1/3 + 1/1 + 1/4 +1/1+1/5…+1/a\_n

Is there an analytic representation of the sum to n terms ? I am guessing the series does not converge - is there a proof ?

Thanks

---

<div class="post-metadata">

**Author:** ![Derleth](https://avatars.discourse-cdn.com/v4/letter/d/b9e5f3/32.png) [@Derleth](https://boards.straightdope.com/u/Derleth)\
**Post date:** [March 23, 2015, 4:32am UTC](https://boards.straightdope.com/t/will-this-series-converge-pi-related/715759/2 "2015-03-23T04:32:21Z")

</div>

(Assuming you remove all terms which would result in 1/0…)

Well, it trivially won’t converge if pi is normal, because then there will be an infinite number of 1/1 terms, so it will keep trending upwards indefinitely.

However, we don’t know if pi is normal.

On the gripping hand, pi being normal is a sufficient but not necessary condition for non-convergence: It is very easy to construct non-normally-distributed infinite sequences which won’t converge if added in the way you propose.

So I don’t see any special reason your sum would converge, and since there’s so many ways for it to diverge, I’m pretty sure it’s going to be divergent.

---

<div class="post-metadata">

**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\
**Post date:** [March 23, 2015, 4:38am UTC](https://boards.straightdope.com/t/will-this-series-converge-pi-related/715759/3 "2015-03-23T04:38:21Z")

</div>

I can think of a way to formulate the summation, but it involves modular arithmetic.  
sum over i=0-\>n { 1/(floor(π\*10[sup]i[/sup]) mod 10) }

And assuming it’s passed through a function that removes 0 values.

I don’t have enough of a background in integer series to be able to analyze this, but I suspect it’s either divergent or (currently) unprovable.

There’s probably a better way to formulate this question in terms of one of the infinite series that evaluate to pi.

---

<div class="post-metadata">

**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\
**Post date:** [March 23, 2015, 4:44am UTC](https://boards.straightdope.com/t/will-this-series-converge-pi-related/715759/4 "2015-03-23T04:44:03Z")

</div>

No. A series 1/9 + 1/9 + 1/9 + … will diverge if it has an infinite number of non-zero terms, so yours will also. And the decimal expansion of pi contains an infinite number of non-zeros; otherwise it would eventually be …0000… forever which happens only for rational numbers.