Aperiodic Monotonic Divergent Series of Integers

VARIETIES

Monotonic divergent series of even numbers (starting from 2)

$\displaystyle \sum_{n\geq1} 2n = 2+4+6+8+10+12+\cdots = -\dfrac{1}{6}$ (unstable)

Monotonic divergent series of multiples of 3 (starting from 3)

$\displaystyle \sum_{n\geq1} 3n = 3+6+9+12+15+18+\cdots = -\dfrac{1}{4}$ (unstable)

Monotonic divergent series of multiples of 4 (starting from 4)

$\displaystyle \sum_{n\geq1} 4n = 4+8+12+16+20+24+\cdots = -\dfrac{1}{3}$ (unstable)

Monotonic divergent series of multiples of 5 (starting from 5)

$\displaystyle \sum_{n\geq1} 5n = 5+10+15+20+25+30+\cdots = -\dfrac{5}{12}$ (unstable)

Monotonic divergent series of oblong numbers (starting from 2)

$\displaystyle \sum_{n\geq1} n(n+1) = 2+6+12+20+30+42+\cdots = -\dfrac{1}{12}$ (unstable)

Monotonic divergent series of odd-indexed bisection of Fibonacci numbers (starting from 1+2)

$\displaystyle \sum_{n\geq1} F_{2n-1} = 1+2+5+13+34+89+\cdots = 0$ (stable)

Monotonic divergent series of even-indexed bisection of Fibonacci numbers (starting from 1+3)

$\displaystyle \sum_{n\geq1} F_{2n} = 1+3+8+21+55+144+\cdots = -1$ (stable)

Monotonic divergent series of odd-indexed bisection of Lucas numbers (starting from 1+4)

$\displaystyle \sum_{n\geq1} L_{2n-1} = 1+4+11+29+76+199+\cdots = -2$ (stable)

Monotonic divergent series of even-indexed bisection of Lucas numbers (starting from 3+7)

$\displaystyle \sum_{n\geq1} L_{2n} = 3+7+18+47+123+322+\cdots = -1$ (stable)

Miscellaneous Series

$1+9+18+27+36+45+54+63+\cdots = \dfrac{19}{4}$ (unstable)

$3+25+50+75+100+125+150+175+\cdots = \dfrac{161}{12}$ (unstable)

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