Terms that vanish, a total that does not

• 1/1 + … + 1/2 is not a whole number ×190
• one number in 1..2 carries the most factors of 2 ×190
• the density is positive at 0.00 ×56
• run 3 of the q = 0.5 series is under its own bound ×24
• 1 golden signs land on F(4) − 1 points ×20
• the histogram bar at -2.47 matches the density ×19
• the histogram bar at 0.07 matches the density ×19
• the vector for λ^1 is λ^1 ×19
• the denominator of H(1) carries 2 to the power ⌊log₂ 1⌋ ×16
• H(2) is not a whole number ×15
• term 2 is under the envelope ×14
• the first 1 terms add to the closed form ×14
• φ's powers close in on whole numbers by a factor 1/φ at 3 ×14
• piece 1 is 0.50 of what was left ×13
• and p³ does not, for p = 5 ×9
• p² divides the numerator of H(4) ×9
• stage 0 has two to the 0 intervals ×8
• and the upper bound on block 0 is the k-th power of 2^(1−p) ×7
• and their lengths add to (2/3) to the 0 ×7
• block 0 sits between its own two bounds ×7
• block 1 reaches a half ×4
• H(4) lies strictly between 2 and 3 ×4
• and that bound is 1/(k log 2)^0.5 ×3
• between 50 and 2000 terms ×2
• the partial sum differs from 1/(1 − 2) by exactly 2 to the power of its number of terms ×2
• the tail of 1/n^1.1 past 100000 is bounded by an integral ×2
• the transform at λ = 0.65 dies away along its own powers ×2
• 2 is first passed at n = 4 ×1
• 3 at n = 11 ×1
• a ratio with even numerator has odd denominator ×1
• and between them they are all 32 strings of 5 bits ×1
• and by the last drawn stage it is already tiny ×1
• and by this stage they already cover most of the interval ×1
• and ends at one ×1
• and it closes on it ×1
• and it is still coming down towards the limit ×1
• and it is the largest power of 2 not above n ×1
• and lists none of them twice ×1
• and no address uses the digit one ×1
• and points of different classes are different points ×1
• and the denominator of the sum carries exactly that many ×1
• and the numerator is odd ×1
• and the remainder is the ratio to the power of the number of pieces ×1
• and they add to what the series says ×1
• and two of the series have ratios climbing to one, one of which converges and one of which does not ×1
• and what has gone is the geometric series it looks like ×1
• and what is left is exactly the tail ×1
• at 1/3 the count is the Cantor set's log 2 / log 3 ×1
• at one half every bar has height a quarter ×1
• at one half it is exactly one ×1
• at one half the distribution function is a straight line ×1
• at p = 0.4 it is large and growing with n ×1
• at p = 1 the variance left after the first tenth is tiny ×1
• at x = 0 every sign is plus and the value is H(digits) ×1
• at x = 1/3 the digits alternate and so do the signs ×1
• below one half most of the interval is empty (λ = 1/3) ×1
• Bertrand: a prime lies between 20/2 and 20 ×1
• between 12 and 40 rationals are covered ×1
• between 2,000 and 40,000 samples ×1
• between 3 and 20 random runs ×1
• between 3 and 7 ternary digits are listed ×1
• between 3 and 9 stages are drawn ×1
• between 8 and 16 binary digits ×1
• between eight and twenty-two terms are drawn ×1
• between four and twelve blocks are drawn ×1
• between ten and forty terms are drawn ×1
• between three and nine translates are drawn ×1
• between three and six exponents between 0.3 and 3 are compared ×1
• between three and six exponents between nought and three are compared ×1
• between three and ten stages are built ×1
• between two and five series the family knows are compared ×1
• between two and four values of λ ×1
• between two and six classes are drawn ×1
• by 4 it is more than ten times thinner than the normal curve ×1
• consecutive partial sums differ by the next term ×1
• each address names the left end of its own interval ×1
• each block leans further out than the one below it ×1
• each class is sampled at between 6 and 30 points ×1
• each rational is inside its own interval ×1
• every matrix drawn has its eigenvalues inside the unit circle ×1
• every partial sum drawn is under the head plus the geometric tail's total ×1
• every partial sum falls short of the limit ×1
• every partial sum sequence reaches (I − A)⁻¹ ×1
• every random run has settled down over its second half ×1
• every such sum has an even denominator ×1
• every triangle is isosceles with its two longest sides equal ×1
• every whole number from 2 up is crossed ×1
• exactly one number up to 12 carries the highest power of 2 ×1
• from one half up the whole interval is used (λ = 1/√2) ×1
• from one half up the whole interval is used (λ = 1/2) ×1
• from one half up the whole interval is used (λ = 1/φ) ×1
• inside the circle the partial sums settle on the inverse ×1
• it starts at nought ×1
• just below 1 every sign is minus ×1
• n runs to between 10 and 30 ×1
• no candidate length puts the union between one and three ×1
• no point is in two classes at once ×1
• no term is used twice ×1
• no two addresses read the same in binary ×1
• no two seeds is a small rational away from another, which would be a typo ×1
• on the circle they stay bounded without settling ×1
• one to four Pisot numbers this family knows ×1
• one to three of the roots 2, 3 and 4 ×1
• only one number up to 20 is a multiple of 19 ×1
• outside it they grow without bound ×1
• so 19 divides the denominator of H(20) exactly once ×1
• so its total is finite ×1
• so the translates of the selection do not overlap ×1
• the 2-adic digits times the denominator give the numerator, to 24 places ×1
• the alternating run is heading for log 2 ×1
• the answer is caught between an even and an odd partial sum ×1
• the bars carry all the probability ×1
• the bound on each block is exactly a half ×1
• the bounding series is geometric exactly when p is above one ×1
• the comparison holds a series from each side of the threshold ×1
• the comparison holds a series that converges and one that does not ×1
• the covered rationals leave no wide gap, so the covering meets every part of the line ×1
• the covering budget is between a hundredth and a half ×1
• the crossings of 2 to between 4 and 11 are drawn ×1
• the denominator of H(12) carries exactly 2^3 ×1
• the density at 0 misses 1/4 by a few millionths ×1
• the density at 2 is 1/8 to double precision ×1
• the density falls away from the centre ×1
• the diagonal matrix's powers only shrink ×1
• the distribution function climbs to one ×1
• the drawn lean is the sum of the drawn steps ×1
• the drawn running total is the partial sum ×1
• the entropy per sign has settled for the golden ratio ×1
• the entropy per sign has settled for the pentanacci ×1
• the entropy per sign has settled for the tetranacci ×1
• the entropy per sign has settled for the tribonacci ×1
• the enumerated bars match the closed form at λ = 1/√2 ×1
• the enumerated bars match the closed form at λ = 1/∛2 ×1
• the envelope dominates the series from some term onwards ×1
• the error is smaller than the last term used ×1
• the exponent is between a half and three ×1
• the first middle removed is between a third and an eighth ×1
• the function is constant across each removed interval ×1
• the function never decreases ×1
• the gap to ln n has settled on γ ×1
• the geometric series' ratio is the same at every term ×1
• the golden dimension is Alexander and Zagier's 0.99571 ×1
• the golden ratio's distribution has dimension below one ×1
• the golden transform holds level at 2πφⁿ ×1
• the halves share bars at λ = 0.6 ×1
• the halves share no bar at λ = 0.4 ×1
• the last term is small ×1
• the length left is the product of what each stage keeps ×1
• the lengths add to less than the budget ×1
• the list holds the rationals it was asked for ×1
• the longest surviving interval shrinks at every stage ×1
• the middle-thirds set of the same depth has far less left ×1
• the n-th root of the size of Aⁿ never falls below the spectral radius ×1
• the neglected tail is under an eighth of a bar ×1
• the neglected tail is under half of a bar ×1
• the other two grow a great deal before they shrink ×1
• the p = 1 row is the harmonic sum, which is a logarithm plus Euler's constant ×1
• the patterns are different ×1
• the patterns disagree only in a leading block ×1
• the pentanacci's distribution has dimension below one ×1
• the pieces and what is left fill the square exactly ×1
• the Pisot values sit just below the line ×1
• the plotted partial sums are the series ×1
• the prime is 2, 3 or 5 ×1
• the ratio is strictly between nothing and one ×1
• the rearranged total settles near the number it was aimed at ×1
• the removed intervals so far have the length the construction gives ×1
• the run is 1 to between 4 and 30 ×1
• the sample variance is near Σ 1/n² = π²/6 ×1
• the series is one the family knows ×1
• the signs are the ones the series has ×1
• the smallest interval is still wide enough to be represented ×1
• the special number is found by powers of two or by a prime ×1
• the staircase is flat over most of its interval ×1
• the sums run to between a thousand and a million terms ×1
• the sums run to between ten thousand and ten million terms ×1
• the surviving length is bounded away from nothing ×1
• the table runs to between 6 and 20 ×1
• the tail is drawn to between 3 and 4.5 ×1
• the target is visibly away from the sum the same terms give in order ×1
• the tetranacci's distribution has dimension below one ×1
• the total is not small ×1
• the triangle has side between 8 and 24 ×1
• the tribonacci's distribution has dimension below one ×1
• the two patterns land on the same point exactly ×1
• the union is no longer than the sum of the pieces ×1
• the view is one the family draws ×1
• there is one address for each surviving interval ×1
• this view is for a series that does converge ×1
• three to ten primes between 3 and 61 ×1
• two different translates never carry the same point of a class twice ×1
• two or three values of λ ×1
• two sign patterns of equal length ×1
• two to four exponents in (0, 1.5] ×1
• two to four values of λ ×1
• what is left and what has gone add to the whole interval ×1
• λ = 1/θ satisfies its relation for the golden ratio ×1
• λ = 1/θ satisfies its relation for the pentanacci ×1
• λ = 1/θ satisfies its relation for the tetranacci ×1
• λ = 1/θ satisfies its relation for the tribonacci ×1
• λ is between 0.2 and 0.9 ×1

The whole library · What the figures prove