A Curious Convergent Series

Harmonic Series

We Know that the Series

Diverges

Proof : If Possible suppose it converges to H , i.e.

H =

Then, H

H

This Contradiction Concludes the Proof

Kempner Series:A modification of harmonic SeriesIn Harmonic series let us remove every term with a 9 in it ( i.e. we remove all the terms 1/9 ,1/19, 1,29 etc.)

So the series that we’re going to deal becomes

Modified SeriesWe form a new series

Where,

Notice that

Similarly

Formal QuestionSo we now state the formal question as:

Consider

Is convergent?

Before we start the proof let us see why our modification to the series is so effective?Consider all integers containing 100 digits. There are of them.

The number of terms that are kept are ,The fraction of terms that we keep is thus

We took lot of terms!!

Convergence of Modified SeriesNotice that, each term of is less than 1 And each term of is less than

And similarly each term of is less than

Thus we have

Using Comparison Test converges!

Estimating Kempner series from aboveComparing with .

The first nine terms of are less than or equal to , so together they’re less than

Next nine terms of are less than or equal to , so together they’re less than

Continuing this process we see that and

Furthermore it can be seen that

Thus we can estimate our modified series from above with

Estimating Kempner series from below

Here the we partition the series differently and estimate each finite sub series from other side.

Define :

We divide the terms of in group of nine, and compare them to the terms in

The first group consists of

Each of them is greater than , so together they’re greater than

Likewise next 9 terms are greater than

Then and

We can see then

So we estimate the series from below by

Thus we’ve found that

Interesting Note Kempner Series also converges if we remove any other digit from or if choose to remove any string of digits like

Forbidden Digit Value of Series

0 23 . 10344 79094 20541 61603

1 16 . 17696 95281 23444 26657

2 19 . 25735 65328 08071 11453

3 20 . 56987 79509 61230 37107

4 21 . 32746 57995 90036 68663

5 21 . 83460 08122 96918 16340

6 22 . 20559 81595 56091 88416

7 22 . 49347 53117 05945 39817

8 22 . 72636 54026 79370 60283

9 22 . 90267 66192 64150 34816

References1. Kempner, A. J. (February 1914). "A Curious Convergent

Series". American Mathematical Monthly

2. Undergrad Thesis of SARAH E. MATZ, THE PENNSYLVANIA STATE UNIVERSITY SCHREYER HONORS COLLEGE.

3. Modified Divergent Series , Blog by Paul Liu

Thank You!!