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A Curious Convergent Series

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 Series.

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A Curious Convergent Series

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  1. A Curious Convergent Series

  2. 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

  3. Kempner Series:A modification of harmonic Series In 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

  4. Modified Series We form a new series Where, Notice that Similarly

  5. Formal Question So 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!!

  6. Convergence of Modified Series Notice 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!

  7. Estimating Kempner series from above Comparing 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

  8. 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

  9. Then and We can see then So we estimate the series from below by Thus we’ve found that

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

  11. References Kempner, A. J. (February 1914). "A Curious Convergent Series". American Mathematical Monthly Undergrad Thesis of SARAH E. MATZ, THE PENNSYLVANIA STATE UNIVERSITY SCHREYER HONORS COLLEGE. Modified Divergent Series , Blog by Paul Liu

  12. Thank You!!

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